Iraduate Texts i Mathematics William A. Adkins Steven H. Weintraub
Algebra An Approach via Module Theory
SpringerVerl...
166 downloads
936 Views
10MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
Iraduate Texts i Mathematics William A. Adkins Steven H. Weintraub
Algebra An Approach via Module Theory
SpringerVerlag
Graduate Texts in Mathematics S. Axler
Springer New York
Berlin Heidelberg Barcelona Hong Kong London
Milan Paris Singapore
Tokyo
136
Editorial Board K.A. Ribet
F.W. Gehring
Graduate Texts in Mathematics I
TAKEUTVZARING. Introduction to
33
34
2
Axiomatic Set Theory. 2nd ed. OxTOBY. Measure and Category. 2nd ed.
3
4
SCHAEFER. Topological Vector Spaces. HILTON/STAMMBACH. A Course in
Homological Algebra. 2nd ed.
HIR.SCH. Differential Topology. SPITZER. Principles of Random Walk.
2nd ed. 35 ALEXANDER/WERMER. Several Complex Variables and Banach Algebras. 3rd ed.
36
KELLEY/NAMIUKA et al. Linear
5
MAC LANE. Categories for the Working Mathematician. 2nd ed.
37
6
HUGHES/PIPER. Projective Planes.
38 GRAUERT/FRI ZSCHE. Several Complex
7
SERRE. A Course in Arithmetic. TAKEUTI/ZARING. Axiomatic Set Theory.
39
HUMPHREYS. Introduction to Lie Algebras
40 KEMENY/SNELIJKNAPP. Denumerable
8
9
and Representation Theory. 10 COHEN. A Course in Simple Homotopy Theory. I I CONWAY. Functions of One Complex Variable 1. 2nd ed. 12 BEALS. Advanced Mathematical Analysis.
Topological Spaces. MONK. Mathematical Logic.
Variables. ARvESON. An Invitation to C*Algebras.
Markov Chains. 2nd ed. APOSTOL. Modular Functions and Dirichlet Series in Number Theory. 2nd ed. 42 SERRE. Linear Representations of Finite 41
Groups.
13
ANDERSON/FULLER. Rings and Categories
43 GILLMAWIERISON. Rings of Continuous
14
of Modules. 2nd ed. GOLuBrrsKY/GuILLEMIN. Stable Mappings
and Their Singularities.
Functions. 44 KENDIG. Elementary Algebraic Geometry. 45 LOEVE. Probability Theory 1. 4th ed.
BERBERIA.v. Lectures in Functional
46
Analysis and Operator Theory.
47
15 16
WINTER. The Structure of Fields.
17
ROSENBLATT. Random Processes. 2nd ed. HALMOS. Measure Theory.
18
HALMOS. A Hilbert Space Problem Book. 2nd ed. 20 HUSEMOLLER. Fibre Bundles. 3rd ed. 21 HUMPHREYS. Linear Algebraic Groups. 22 BARNES/MACK. An Algebraic Introduction 19
to Mathematical Logic. 23 GREUB. Linear Algebra. 4th ed. 24 HOLMES. Geometric Functional Analysis and Its Applications. 25 26
27 28 29
HEwrrr/STROMSERG. Real and Abstract
Analysis. MANES. Algebraic Theories. KELLEY. General Topology. ZARISKUSAMUEL. Commutative Algebra.
56 MASSEY. Algebraic Topology: An
Introduction.
Vol.l.
57
ZARIsKi/SAMuEL. Commutative Algebra. Vo1.1I.
58
30 JACOBSON. Lectures in Abstract Algebra I. Basic Concepts. 31 JACOBSON. Lectures in Abstract Algebra 11. Linear Algebra. 32
LOEvE. Probability Theory 11. 4th ed. MOISE. Geometric Topology in Dimensions 2 and 3. 48 SACHS/WU. General Relativity for Mathematicians. 49 GRUENBERO/WEIR. Linear Geometry. 2nd ed. 50 EDWARDS. Fermat's Last Theorem. 51 KLINGENBERG. A Course in Differential Geometry. 52 HARTSHORNE. Algebraic Geometry. 53 MANIN. A Course in Mathematical Logic. 54 GRAVER/WATKINS. Combinatorics with Emphasis on the Theory of Graphs. 55 BROwN/PEARCY. Introduction to Operator Theory 1: Elements of Functional Analysis.
JACOBSON. Lectures in Abstract Algebra
III. Theory of Fields and Galois Theory.
CROWELL/FOX. Introduction to Knot Theory. KOBLITz padic Numbers, padic
Analysis. and ZetaFunctions. 2nd ed. 59
LANG. Cyclotomic Fields.
60 ARNOLD. Mathematical Methods in 61
Classical Mechanics. 2nd ed. WHITEHEAD. Elements of Homotopy Theory.
(continued after index)
William A. Adkins
Steven H. Weintraub
Algebra An Approach via Module Theory
Springer
William A. Adkins Steven H. Weintraub Department of Mathematics Louisiana State University Baton Rouge, LA 70803 USA Editorial Board S. Axler Mathematics Department San Francisco State University San Francisco. CA 94132 USA
F.W. Gehring Mathematics Department
East Hall University of Michigan Ann Arbor, MI 48109 USA
K.A. Ribet
Department of Mathematics University of California at Berkeley Berkeley, CA 947203840 USA
Mathematics Subject Classifications: 1201, 1301, 1501, 1601, 2001 Library of Congress CataloginginPublication Data Adkins, William A. Algebra: an approach via module theory/William A. Adkins, Steven H. Weintraub. p. cm.  (Graduate texts in mathematics; 136) Includes bibliographical references and indexes.. ISBN 0387978399.  ISBN 3540978399 1. Algebra. 2. Modules (Algebra) 1. Weintraub, Steven H. If. Title. Ill. Series. QA 154.A33 1992 512'.4  dc20
9211951
Printed on acidfree paper. J 1992 SpringerVerlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (SpringerVerlag New York, Inc., 175 Fifth Avenue, New
York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafer developed is forbidden. The use of general descriptive names, trade names, trademarks. etc., in this publication, even if
the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Production managed by Francine Sikorski: manufacturing supervised by Jacqui Ashri. Photocomposed copy prepared using TeX. Printed and bound by R.R. Donnelley and Sons, Harrisonburg, VA. Printed in the United States of America. 9 8 7 6 5 4 3 2 (Corrected second printing, 1999)
ISBN 0387978399 SpringerVerlag New York Berlin Heidelberg ISBN 3540978399 SpringerVerlag Berlin Heidelberg New York SPIN 10667846
Preface
This book is designed as a text for a firstyear graduate algebra course. As necessary background we would consider a good undergraduate linear algebra course. An undergraduate abstract algebra course, while helpful, is not necessary (and so an adventurous undergraduate might learn some algebra from this book). Perhaps the principal distinguishing feature of this book is its point of view. Many textbooks tend to be encyclopedic. We have tried to write one that is thematic, with a consistent point of view. The theme, as indicated by our title, is that of modules (though our intention has not been to write a textbook purely on module theory). We begin with some group and ring theory, to set the stage, and then, in the heart of the book, develop module theory. Having developed it, we present some of its applications: canonical forms for linear transformations, bilinear forms, and group representations. Why modules? The answer is that they are a basic unifying concept
in mathematics. The reader is probably already familiar with the basic role that vector spaces play in mathematics, and modules are a generalization of vector spaces. (To be precise, modules are to rings as vector spaces are to fields.) In particular, both abelian groups and vector spaces with a linear transformation are examples of modules, and we stress the analogy between the twothe basic structure theorems in each of these areas are special cases of the structure theorem of finitely generated modules over a principal ideal domain (PID). As well, our last chapter is devoted to the representation theory of a group G over a field F, this being an important
and beautiful topic, and we approach it from the point of view of such a representation being an F(G)module. On the one hand, this approach makes it very clear what is going on, and on the other hand, this application shows the power of the general theory we develop. We have heard the joke that the typical theorem in mathematics states that something you do not understand is equal to something else you can
not compute. In that sense we have tried to make this book atypical. It has been our philosophy while writing this book to provide proofs with a
vi
Preface
maximum of insight and a minimum of computation, in order to promote understanding. However, since in practice it is necessary to be able to compute as well, we have included extensive material on computations. (For
example, in our entire development in Chapter 4 of canonical forms for linear transformations we only have to compute one determinant, that of a companion matrix. But then Chapter 5 is almost entirely dedicated to computational methods for modules over a PID, showing how to find canonical forms and characteristic polynomials. As a second example, we derive the basic results about complex representations of finite groups in Section 8.3, without mentioning the word character, but then devote Section 8.4 to characters and how to use them.)
Here is a more detailed listing of the contents of the book, with emphasis on its novel features: Chapter 1 is an introduction to (or review of) group theory, including a discussion of semidirect products. Chapter 2 is an introduction to ring theory, covering a variety of standard topics. In Chapter 3 we develop basic module theory. This chapter culminates
in the structure theorem for finitely generated modules over a PID. (We then specialize to obtain the basic structure theorem for finitely generated Abelian groups.) We feel that our proof of this theorem is a particularly insightful one. (Note that in considering free modules we do not assume the corresponding results for vector spaces to be already known.) Noteworthy along the way is our introduction and use of the language of homological algebra and our discussion of free and projective modules.
We begin Chapter 4 with a treatment of basic topics in linear algebra. In principle, this should be a review, but we are careful to develop as much of the theory as possible over a commutative ring (usually a PID) rather than just restricting ourselves to a field. The matrix representation for module homomorphisms is even developed for modules over noncommutative rings, since this is needed for applications to Wedderburn's theorem in Chapter 7. This chapter culminates in the derivation of canonical forms
(the rational canonical form, the (generalized) Jordan canonical form) for linear transformations. Here is one place where the module theory shows its worth. By regarding a vector space V over a field F, with a linear transformation T, as an F[X)module (with X acting by T), these canonical forms are immediate consequences of the structure theorem for finitely generated torsion modules over a PID. We also derive the important special case of the real Jordan canonical form, and end the chapter by deriving the spectral theorem. Chapter 5 is a computational chapter, showing how to obtain effectively (in so far as is possible) the canonical forms of Chapter 4 in concrete cases. Along the way, we introduce the Smith and Hermite canonical forms as well.
Preface
vii
This chapter also has Dixon's proof of a criterion for similarity of matrices based solely on rank computations.
In Chapter 6 we discuss duality and investigate bilinear, sesquilinear, and quadratic forms, with the assistance of module theory, obtaining complete results in a number of important special cases. Among these are the cases of skewsymmetric forms over a PID, sesquilinear (Hermitian) forms over the complex numbers, and bilinear and quadratic forms over the real numbers, over finite fields of odd characteristic, and over the field with two elements (where the Arf invariant enters in the case of quadratic forms).
Chapter 7 has two sections. The first discusses semisimple rings and modules (deriving Wedderburn's theorem), and the second develops some multilinear algebra. Our results in both of these sections are crucial for Chapter 8. Our final chapter, Chapter 8, is the capstone of the book, dealing with group representations mostly, though not entirely, in the semisimple case.
Although perhaps not the most usual of topics in a firstyear graduate course, it is a beautiful and important part of mathematics. We view a representation of a group G over a field F as an F(G)module, and so this chapter applies (or illustrates) much of the material we have developed in this book. Particularly noteworthy is our treatment of induced representations. Many authors define them more or less ad hoc, perhaps mentioning as an aside that they are tensor products. We define them as tensor products and stick to that point of view (though we provide a recognition principle not involving tensor products), so that, for example, Frobenius reciprocity merely becomes a special case of adjoint associativity of Hom and tensor product. The interdependence of the chapters is as follows: 1
I.
2 1
3 I.
4.14.3 1
4.44.6
F61
7
1
J.
5
8
viii
Preface
We should mention that there is one subject we do not treat. We do not discuss any field theory in this book. In fact, in writing this book we were careful to avoid requiring any knowledge of field theory or algebraic number theory as a prerequisite. We use standard set theoretic notation. For the convenience of the reader, we have provided a very brief introduction to equivalence relations and Zorn's lemma in an appendix. In addition, we provide an index of notation, with a reference given of the first occurrence of the symbol. We have used a conventional decimal numbering system. Thus a reference to Theorem 4.6.23 refers to item number 23 in Section 6 of Chapter 4, which happens to be a theorem. Within a given chapter, the chapter reference is deleted.
The symbol O is used to denote the end of a proof; the end of proof symbol 0 with a blank line is used to indicate that the proof is immediate from the preceding discussion or result. The material presented in this book is for the most part quite standard. We have thus not attempted to provide references for most results. The bibliography at the end is a collection of standard works on algebra. We would like to thank the editors of SpringerVerlag for allowing us the opportunity, during the process of preparing a second printing, to correct a number of errors which appeared in the first printing of this book. Moreover, we extend our thanks to our colleagues and those readers who have taken the initiative to inform us of the errors they have found. Michal Jastrzebski and Lyle Ramshaw, in particular, have been most helpful in pointing out mistakes and ambiguities. Baton Rouge, Louisiana
William A. Adkins Steven H. Weintraub
Contents
Preface
Chapter 1
.
.
.
.
.
Groups
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
V
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
.
.
1
.
.
6
.
.
.
.
.
.
.
.
.
.
.
.
22 28 34 39 45
.
.
.
49
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
49 58 68 72 79 92 98 107
1.1 Definitions and Examples . . . . . . . . . . . . 1.2 Subgroups and Cosets . . . . . . . . . . . . . . 1.3 Normal Subgroups, Isomorphism Theorems, and Automorphism Groups . . . . . . . . . . . 1.4 Permutation Representations and the Sylow Theorems 1.5 The Symmetric Group and Symmetry Groups . . . 1.6 Direct and Semidirect Products . . . . . . . . . . 1.7 Groups of Low Order . . . . . . . . . . . . . . 1.8 Exercises . . . . . . . . . . . . . . . . . . . .
Chapter 2
Rings
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2.1 Definitions and Examples . . . . . . . . . . . 2.2 Ideals, Quotient Rings, and Isomorphism Theorems 2.3 Quotient Fields and Localization . . . . . . . . 2.4 Polynomial Rings . . . . . . . . . . . . . . . 2.5 Principal Ideal Domains and Euclidean Domains . 2.6 Unique Factorization Domains . . . . . . . . 2.7 Exercises . . . . . . . . . . . . . . . . . . . .
Chapter 3
Modules and Vector Spaces
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3.1 Definitions and Examples . . . . . 3.2 Submodules and Quotient Modules . . 3.3 Direct Sums, Exact Sequences, and Horn 3.4 Free Modules . . . . . . . . . . . . 3.5 Projective Modules . . . . . . . . . 3.6 Free Modules over a PID . . . . . . 3.7 Finitely Generated Modules over PIDs . 3.8 Complemented Submodules . . . . 3.9 Exercises . . . . . . . . . . . . . .
.
15
107 112 118 128 136 142 156 171 174
x
Contents
Chapter 4
Linear Algebra
.
182
4.1 Matrix Algebra . . . . . . . . . . . . . . . . . . 4.2 Determinants and Linear Equations . . . . . . . . . . 4.3 Matrix Representation of Homomorphisms . . . . . . . 4.4 Canonical Form Theory . . . . . . . . . . . . . . . 4.5 Computational Examples . . . . . . . . . . . . . . 4.6 Inner Product Spaces and Normal Linear Transformations 4.7 Exercises . . . . . . . . . . . . . . . . . . . . .
182 194 214 231 257 269 278
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Chapter 5
Matrices over PIDs
.
.
.
5.1 Equivalence and Similarity . . 5.2 Hermite Normal Form . . . . 5.3 Smith Normal Form . . . . . 5.4 Computational Examples . . 5.5 A Rank Criterion for Similarity 5.6 Exercises . . . . . . . . . .
Chapter 6
.
.
Chapter 7
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
289 296 307 319 328 337
.
.
.
.
.
.
.
.
.
.
.
341
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
341 350 376 391
.
.
.
.
.
.
.
.
.
.
.
.
395
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
395 412 434
.
.
.
.
.
.
.
438
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Bilinear and Quadratic Forms
6.1 Duality . . . . . . . . . . . 6.2 Bilinear and Sesquilinear Forms 6.3 Quadratic Forms . . . . . 6.4 Exercises . . . . . . . . . Topics in Module Theory
7.1 Simple and Semisimple Rings and Modules 7.2 Multilinear Algebra . . . . . . . . . 7.3 Exercises . . . . . . . . . . . . . . .
.
Chapter 8
Group Representations
.
.
.
.
.
.
.
8.1 Examples and General Results . . . . . 8.2 Representations of Abelian Groups . . . . 8.3 Decomposition of the Regular Representation 8.4 Characters . . . . . . . . . . . . . 8.5 Induced Representations . . . . . . . . . 8.6 Permutation Representations . . . . . . . 8.7 Concluding Remarks . . . . . . . . . 8.8 Exercises . . . . . . . . . . . . . . . .
.
.
.
.
Appendix . . . . . Bibliography . . . Index of Notation Index of Terminology
289
.
438 451 453 462 479 496 503 505
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
507 510
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
511
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
517
Chapter 1
Groups
In this chapter we introduce groups and prove some of the basic theorems in group theory. One of these, the structure theorem for finitely generated abelian groups, we do not prove here but instead derive it as a corollary of the more general structure theorem for finitely generated modules over a PID (see Theorem 3.7.22).
1.1 Definitions and Examples (1.1) Definition. A group is a set G together with a binary operation
:GxG'G satisfying the following three conditions:
(a) a (b c) = (a b) c for all a, b, c E G. (Associativity) (b)
There exists an element e E G such that a e = e a = a for all a E G. (Existence of an identity element)
(c) For each a E G there exists a b E G such that a b = b a = e. (Existence
of an inverse for each a E G)
It is customary in working with binary operations to write a b rather than .(a, b). Moreover, when the binary operation defines a group structure
on a set G then it is traditional to write the group operation as ab. One exception to this convention occurs when the group G is abelian, i.e., if ab = ba for all a, b E G. If the group G is abelian then the group operation is commonly written additively, i.e., one writes a + b rather than ab. This convention is not rigidly followed; for example, one does not suddenly
switch to additive notation when dealing with a group that is a subset of a group written multiplicatively. However, when dealing specifically with abelian groups the additive convention is common. Also, when dealing with abelian groups the identity is commonly written e = 0, in conformity with
2
Chapter 1. Groups
the additive notation. In this chapter, we will write e for the identity of general groups, i.e., those written multiplicatively, but when we study group representation theory in Chapter 8, we will switch to 1 as the identity for multiplicatively written groups. To present some examples of groups we must give the set G and the operation : G x C  G and then check that this operation satisfies (a), (b), and (c) of Definition 1.1. For most of the following examples, the fact that the operation satisfies (a), (b), and (c) follows from properties of the
various number systems with which you should be quite familiar. Thus details of the verification of the axioms are generally left to the reader. (1.2) Examples.
(1) The set Z of integers with the operation being ordinary addition of integers is a group with identity e = 0, and the inverse of m E Z is m. Similarly, we obtain the additive group Q of rational numbers, R of real numbers, and C of complex numbers. (2) The set Q' of nonzero rational numbers with the operation of ordinary multiplication is a group with identity e = 1, and the inverse of a E Q' is 1/a. Q' is abelian, but this is one example of an abelian group that is not normally written with additive notation. Similarly, there are the abelian groups R' of nonzero real numbers and C* of nonzero complex numbers. (3) The set Z = {0, 1, ... , n 1} with the operation of addition modulo n is a group with identity 0, and the inverse of x E Z is nx. Recall that addition modulo n is defined as follows. If x, y E Z,, take x + y E Z
and divide by n to get x + y = qn + r where 0 < r < n. Then define x + y (mod n) to be r. (4) The set U,, of complex nt^ roots of unity, i.e., U = {exp((2k7ri)/n) 0 < k < n  11 with the operation of multiplication of complex numbers is a group with the identity e = 1 = exp(0), and the inverse of exp((2kai)/n) is exp((2(n  k)7ri)/n). (5) Let Z;, = {m : 1 < m < n and m is relatively prime to n}. Under the operation of multiplication modulo n, Z;, is a group with identity 1. Details of the verification are left as an exercise. (6)
If X is a set let Sx be the set of all bijective functions f : X
X. Recall that a function is bijective if it is onetoone and onto. Functional
composition gives a binary operation on Sx and with this operation it becomes a group. Sx is called the group of permutations of X or the symmetric group on X. If X = {1, 2, ..., n} then the symmetric
group on X is usually denoted S and an element a of S,, can be conveniently indicated by a 2 x n matrix a

1
2
a(1)
a(2)
n ...
a(n)
1.1 Definitions and Examples
3
where the entry in the second row under k is the image a(k) of k under the function a. To conform with the conventions of functional composition, the product afl will be read from right to left, i.e., first do Q and then do a. For example, 1
(3
2 2
3 4
4
1) (3
4
3
4
1
2)
 (4
1
3
4
3
2)
(7) Let GL(n, R) denote the set of n x n invertible matrices with real entries. Then GL(n, R) is a group under matrix multiplication. Let SL(n, R) = IT E GL(n, R) : detT = 11. Then SL(n,R) is a group under matrix multiplication. (In this example, we are assuming familiarity with basic properties of matrix multiplication and determinants. See Chapter 4 for details.) GL(n, R) (respectively, SL(n, R)) is known as the general linear group (respectively, special linear group) of degree n over R. (8) If X is a set let P(X) denote the power set of X, i.e., P(X) is the set of all subsets of X. Define a product on P(X) by the formula AD B = (A \ B) U (B \ A). A D B is called the symmetric difference of A and B. It is a straightforward exercise to verify the associative law for the symmetric difference. Also note that AAA = 0 and 0/IA = AL0 = A. Thus P(X) with the symmetric difference operation is a group with 0 as identity and every element as its own inverse. Note that P(X) is an abelian group. (9) Let C(R) be the set of continuous realvalued functions defined on R and let D(R) be the set of differentiable realvalued functions defined on R. Then C(R) and D(R) are groups under the operation of function addition. One way to explicitly describe a group with only finitely many elements is to give a table listing the multiplications. For example the group {1, 1} has the multiplication table
1
whereas the following table
1
Chapter 1. Groups
4
e
a
b
c
e
e
a
b
c
a
a
e
c
b
b
b
c
e
a
c
c
b
a
e
is the table of a group called the Klein 4group. Note that in these tables each entry of the group appears exactly once in each row and column. Also the multiplication is read from left to right; that is, the entry at the intersection of the row headed by a and the column headed by Q is the product aQ. Such a table is called a Cayley diagram of the group. They are sometimes useful for an explicit listing of the multiplication in small groups. The following result collects some elementary properties of a group:
(1.3) Proposition. Let G be a group. The identity e of G is unique. The inverse b of a E G is unique. We denote it by al. (3) (a')' = a for all a E G and = bla1 for all a, b E G. (4) If a, b E G the equations ax = b and ya = b each have unique solutions in G. (5) If a, b, c E G then ab = ac implies that b = c and ab = cb implies that (1) (2)
(ab)l
a=c.
Proof. (1) Suppose e' is also an identity. Then e' = e'e = e.
(2) Suppose ab = ba = e and all = b'a = e. Then b = eb = (b'a)b = b'(ab) = b'e = b', so inverses are unique. (3) a(al) = (a1)a = e, so (al)l = a. Also (ab)(blal) = a(bb')al
blal.
=
asl
= e and similarly (blal)(ab) = e. Thus (ab)l =
(4) x = a'b solves ax = b and y =
ba' solves
ya = b, and any
solution must be the given one as one sees by multiplication on the left or
right by al. (5) If ab = ac then b = al(ab) = al(ac) = c.
0
The results in part (5) of Proposition 1.3 are known as the cancellation laws for a group. The associative law for a group G shows that a product of the elements a, b, c of G can be written unambiguously as abc. Since the multiplication is binary, what this means is that any two ways of multiplying a, b, and c (so that the order of occurrence in the product is the given order) produces the same element of G. With three elements there are only two choices for
multiplication, that is, (ab)c and a(bc), and the law of associativity says
1.1 Definitions and Examples
5
that these are the same element of G. If there are n elements of G then the law of associativity combined with induction shows that we can write a1a2 . an unambiguously, i.e., it is not necessary to include parentheses to indicate which sequence of binary multiplications occurred to arrive at an element of G involving all of the a1. This is the content of the next proposition.
(1.4) Proposition. Any two ways of multiplying the elements a1, a2, ..., an in a group G in the order given (i.e., removal of all parentheses produces the juxtaposition ala2 an) produces the same element of G.
Proof. If n = 3 the result is clear from the associative law in G.
Let n > 3 and consider two elements g and h obtained as products of a1, a2, ..., an in the given order. Writing g and h in terms of the last multiplication used to obtain them gives g = (a1 ... a:) (aj+1 ... an) and h =
Since i and j are less than n, the induction hypothesis implies that the products al a;, a;+1 an, al aj, and a,+1. . an are unambiguously defined elements in G. Without loss of generality we may assume that i < j.
If i = j then g = h and we are done. Thus assume that i < j. Then, by the induction hypothesis, parentheses can be rearranged so that g = (al ... a;)((ai+l ... ai)(a,+1 ... an)) and
h = ((a1 ...
a:)(ai+1...
ai))(ai+1... an).
Letting A = (a1 ...a;), B = (a;+1 ...a,), and C = (a,+1 ... an) the induction hypothesis implies that A, B, and C are unambiguously defined elements of G. Then
g = A(BC) = (AB)C = h and the proposition follows by the principle of induction.
0
Since products of n elements of G are unambiguous once the order has been specified, we will write a1a2 an for such a product, without any specification of parentheses. Note that the only property of a group used in Proposition 1.4 is the associative property. Therefore, Proposition 1.4 is valid for any associative binary operation. We will use this fact to be able to write unambiguous multiplication of elements of a ring in later chapters. A an is 11,n_1 a;. If a; = a for all i then 11ni=1 a is convenient notation for a1 denoted an and called the nth power of a. Negative powers of a are defined
Chapter 1. Groups
6
by a" = (a1)n where n > 0, and we set a° = e. With these notations the standard rules for exponents are valid. (1.5) Proposition. If G is a group and a E G then
(1) aman = am+n , and (2) (am)n = amn for all integers m and n. Proof. Part (1) follows from Proposition 1.4 while part (2) is an easy exercise using induction.
1.2 Subgroups and Cosets Let G be a group and let H C G be a subset. H is called a subgroup of G if H together with the binary operation of G is a group. The first thing to note is that this requires that H be closed under the multiplication of G, that is, ab is in H whenever a and b are in H. This is no more than the statement that the multiplication on G is defined on H. Furthermore, if H is a subgroup of G then H has an identity e' and G has an identity e. Then e'e = e' since e is the identity of G and e'e' = e' since e' is the identity of H. Thus e'e = e'e' and left cancellation of e' (in the group G) gives e = e'. Therefore, the identity of G is also the identity of any subgroup H of G. Also, if a E H then the inverse of a as an element of H is the same as the inverse of a as an element of G since the inverse of an element is the unique
solution to the equations ax = e = xa.
(2.1) Proposition. Let G be a group and let H be a nonempty subset of G. Then H is a subgroup if and only if the following two conditions are satisfied. (1) If a, b E H then ab E H. (2) If a E H then a'1 E H.
Proof. If H is a subgroup then (1) and (2) are satisfied as was observed in the previous paragraph. If (1) and (2) are satisfied and a E H then a1 E H
by (2) and e = aa1 E H by (1). Thus conditions (a), (b), and (c) in the definition of a group are satisfied for H, and hence H is a subgroup of G. 0 (2.2) Remarks. (1) Conditions (1) and (2) of Proposition 2.1 can be replaced by the following single condition.
(1)' If a, b E H then ab1 E H.
1.2 Subgroups and Cosets
7
Indeed, if (1)' is satisfied then whenever a E H it follows that e = as1 E H and then a1 = ea1 E H. Thus a E H implies that a1 E H. Also, if a, b E H then b'1 E H so that ab = implies (1) and (2). The other implication is clear.
a(b1)1
E H. Therefore, (1)'
(2) If H is finite then only condition (1) of Proposition 2.1 is necessary to ensure that H is a subgroup of G. To see this suppose that H is a finite set and suppose that a, b E H implies that ab E H. We need to show that a1 E H for every a E H. Thus let a E H and let Ta : H + H be defined by Ta(b) = ab. Our hypothesis implies that Ta(H) C H. If Ta(b) = T0(c) then ab = ac and left cancellation in the group G (Proposition 1.3 (5)) shows that b = c. Hence T. is an injective map and, since H is assumed to be finite, it follows that T. is bijective, so the equation ax = c is solvable in H for any choice of c E H. Taking c = a shows that e E H and then taking c = e shows that a1 E H. Therefore, condition (2) of Proposition 2.1 is satisfied and H is a subgroup of G. (3) If G is an abelian group with the additive notation, then H C G is a subgroup if and only if a  b E H whenever a, b E H.
(2.3) Proposition. Let I be an index set and let Hi be a subgroup of G for each i E 1. Then H = fiE, Hi is a subgroup of G. Proof. If a, b E H then a, b E Hi for all i E I. Thus ab1 E Hi for all i E I. Hence ab1 E H and H is a subgroup by Remark 2.2 (1). 0
(2.4) Definition. Let G and H be groups and let f : G ' H be a function. Then f is a group homomorphism if f (ab) = f (a) f (b) for all a, b E G. A group isomorphism is an invertible group homomorphism. If f is a group homomorphism, let
Ker(f) = {aEG : f (a) = e} and
Im(f)={hEH:h=f(a) for some aEG}. Ker(f) is the kernel of the homomorphism f and Im(f) is the image of f. It is easy to check that f is invertible as a group homomorphism if and
only if it is invertible as a function between sets, i.e., if and only if it is bijective.
(2.5) Proposition. Let f : G  H be a group homomorphism. Then Ker(f) and lm(f) are subgroups of G and H respectively.
Proof. First note that f (e) = f(ee) = f (e) f (e), so by cancellation in H we conclude that f (e) = e. Then e = f (e) = f (aa1) = f (a) f (a1) for all a E G. Thus f (a1) = f (a)1 for all a E G. Now let a, b E Ker(f ). Then f (ab1) = f (a) f (b1) = f (a) f (b)1 = ee1 = e, so ab1 E Ker(f)
Chapter 1. Groups
8
and Ker(f) is a subgroup of G. Similarly, if f (a), f (b) E Im(f) then f (a) f (b)1 = f (ab1) E Im(f ), so Im(f) is a subgroup of H. 0 (2.6) Definition. Let S be a subset of a group G. Then (S) denotes the intersection of all subgroups of G that contain S. The subgroup (S) is called
the subgroup generated by S. If S is finite and G = (S) we say that G is finitely generated. If S = {a} has only one element and G = (S) then we say that G is a cyclic group. (2.7) Proposition. Let S be a nonempty subset of a group G. Then
(S)={a,a2...an:nENanda;or aj1ESfor 1 j. The subset T(n, R) C GL(n, R) of invertible upper triangular matrices is a subgroup of GL(n, R). (9) If G is a group let Z(G), called the center of G, be defined by
Z(G)={aEG:ab=ba forallbEG}. Then Z(G) is a subgroup of G.
1.2 Subgroups and Cosets
9
(10) If G is a group and x E G, then the centralizer of x is the subset C(x) of G defined by
C(x)={aEG:ax=xa}. C(x) is a subgroup of G and C(x) = G if and only if x E Z(G). Also note that C(x) always contains the subgroup (x) generated by x. (11) If G is a group and a, b E G, then [a, b] = a' b' ab is called the commutator of a and b. The subgroup G' generated by all the commutators
of elements of G is called the commutator subgroup of G. Another common notation for the commutator subgroup is [G, Cl. See Exercise 22 for some properties of the commutator subgroup. (12) A convenient way to describe some groups is by giving generators and relations. Rather than giving formal definitions we shall be content to illustrate the method with two examples of groups commonly expressed
by generators and relations. For the first, the quaternion group is a group with 8 elements. There are two generators a and b subject to the three relations (and no others): a4 = e;
blab = a1.
b2 = a2;
We leave it for the reader to check that Q = {e, a, a2, a3, b, ab, a2b, a3b}.
For a concrete description of Q as a subgroup of GL(2, C), see Exercise 24.
(13) As our second example of a group expressed by generators and relations, the dihedral group of order 2n, denoted D2n, is a group generated by two elements x and y subject to the three relations (and no others): yxy1 = x1. xn = e; y2 = e; Again, we leave it as an exercise to check that D2n = {e, x, x2, ... ,xn1, ... y, yx, Y X
,yxn1
Thus, D2n has 2n elements. The dihedral group will be presented as a group of symmetries in Section 1.6, and it will be studied in detail from the point of view of representation theory in Chapter 8. (2.9) Definition. The order of G, denoted IGI, is the cardinality of the set G. The order of an element a E G, denoted o(a) is the order of the subgroup
generated by a. (In general, IXI will denote the cardinality of the set X, with IXI = oo used to indicate an infinite set.) (2.10) Lemma. Let G be a group and a E G. Then
(1) o(a) = oo if and only if an 34 e for any n > 0.
10
(2)
Chapter 1. Groups
If o(a) < oo, then o(a) is the smallest positive integer n such that
an = e. (3) ak = e if and only if o(a) I k. Proof. (1) If an34 e for any n > 0, then a'
a' for any r#ssince a'=a'
implies ar' = e = a'r, and if r 54 s, then r  s > 0 or s  r > 0, which is excluded by our hypothesis. Thus, if an # e for n > 0, then I (a) = oo, so o(a) = oo. If an = e then let a'n be any element of (a). Writing m = qn + r
where 0 < r < n we see that a' = anq+r = angar = (an)gar = egar = Thus (a) = {e, a, a2, ..., an1} and o(a) < n < oo.
ar.
(2) By part (1), if o(a) < oo then there is an n > 0 such that a" = e and for each such n the argument in (1) shows that (a) = {e, a, ..., an1}. If we choose n as the smallest positive integer such that an = e then we claim that the powers a' are all distinct for 0 < i < n  1. Suppose that a' = a3
for 0 < i < j < n  1. Then aj' = e and 0 < j  i < n, contradicting the choice of n. Thus o(a) = n = smallest positive integer such that an = e.
(3) Assume that ak = e, let n = o(a), and write k = nq + r where 0 < r < n. Then e = ak = anq+r = angar = ar. Part (2) shows that we
must have r=0sothat k=nq.
0
We will now characterize all subgroups of cyclic groups. We start with the group Z.
(2.11) Theorem. If H is a subgroup of Z then H consists of all the multiples of a fixed integer m, i.e., H = (m).
Proof. If H = {0} we are done. Otherwise H contains a positive integer
since H contains both n and n whenever it contains n. Let m be the least positive integer in H. Then we claim that H = {km: k E Z} = (m). Indeed, let n E H. Then write n = qm + r where 0 < r < m. Since n E H and m E H, it follows that r = n  qm E H because H is a subgroup of Z. But 0 < r < m so the choice of m forces r = 0, otherwise r is a smaller positive integer in H than m. Hence n = qrn so that every element of H is a multiple of m, as required. 0 We now determine all subgroups of a cyclic group G. Assume that G = (a) and let H be a subgroup of G such that H # {e}. If H contains a power
a' with a negative exponent then it also contains the inverse a', which is a positive power of a. Arguing as in Theorem 2.11, let m be the smallest positive integer such that a'n E H. Let a' be an arbitrary element of H and write s = qm + r where 0 < r < m. Then ar = a'q, = a'(am)q E H
since a' and a' are in H. Thus we must have r = 0 since r < m and in is the smallest positive integer with a'n E H. Therefore, s = qm and a' = (a')q so that all elements of H are powers of a". If a is of finite order n so that an = e then n must be divisible by
m because e = an E H so that n = qm for some q. In this case, H =
1.2 Subgroups and Cosets
11
{e,am,a2m' ,a(Q1)m}. Therefore, JHI = q = n/m. However, if the order of a is infinite, then H = {e, a±m, a±2m, ... } = (am) is also infinite cyclic. Thus we have proved the following result. (2.12) Theorem Any subgroup H of a cyclic group G = (a) is cyclic. Moreover, either H = (e) or H = (am) where m is the smallest positive power of a that is in H. If G is infinite then m is arbitrary and H is infinite cyclic.
If IGI = n then m In and JHI = n/m. If m is any factor of n then there is exactly one subgroup H of G of order n/m, namely, H = (a'). The above theorem gives a complete description of cyclic groups and their subgroups. From this description, it is easy to see that any two cyclic groups of order n are isomorphic, as well as any two infinite cyclic groups are isomorphic. Indeed, if G = (a) and H = (b) where IGI = IHI = n then
define f : G  H by f (am) = b'° for all m. One checks that f is a group isomorphism. In particular, every cyclic group of order n is isomorphic to the additive group Zn of integers modulo n (see Example 1.2 (3)), and any infinite cyclic group is isomorphic to the additive group Z. (2.13) Definition. Let G be a group and H a subgroup. For a fixed element a E G we define two subsets of G: (1) The left coset of H in G determined by a is the set aH = {ah : h E H}.
The element a is called a representative of the left coset aH. (2) The right coset of H in G determined by a is the set Ha = {ha : h E
H}. The element a is called a representative of the right coset Ha. Remark. Unfortunately, there is no unanimity on this definition in the mathematical world. Some authors define left and right cosets as we do; others have the definitions reversed. A given left or right coset of H can have many different representatives. The following lemma gives a criterion for two elements to represent the same coset.
(2.14) Lemma. Let H be a subgroup of G and let a, b E G. Then
(1) aH = bH if and only if a1b E H, and (2) Ha = Hb if and only if ab1 E H. Proof. We give the proof of (1). Suppose a1b E H and let b = ah for some h E H. Then bh' = a(hh') for all h' E H and ah1 = (ah)(h1h1) = b(h1h1)
for all h1 E H. Thus aH = W. Conversely, suppose aH = W. Then 0
b = be = ah for some h E H. Therefore, a1b = h E H.
12
Chapter 1. Groups
(2.15) Theorem. Let H be a subgroup of G. Then the left cosets (right cosecs)
of H form a partition of G. Proof. Define a relation L on G by setting a ^'L b if and only if a'b E H. Note that
(1) a'.La, (2) a ^'L b implies b L a (since alb E H implies that b'a = (a'b)l E H), and (3) a .L b and b L c implies a L c. Thus, L is an equivalence relation on G and the equivalence classes of L, denoted [a]L, partition G. (See the appendix.) That is, the equivalence classes [a]L and [b]L are identical or they do not intersect. But
[a]L={bEG:aLb}
={bEG:a'bEH} = {b e C: b= ah for some hE H} = aH. Thus, the left cosets of H partition G and similarly for the right cosets.
The function ¢a : H , aH defined by ma(h) = ah is bijective by the left cancellation property. Thus, every left coset of H has the same cardinality as H, i.e., (aHI =IHI for every a E G. Similarly, by the right cancellation law the function tha(h) = ha from H to Ha is bijective so that every right coset of H also has the same cardinality as H. In particular, all right and left cosets of H have the same cardinality, namely, that of H itself.
(2.16) Definition. If H is a subgroup of G we define the index of H in G, denoted [G : H], to be the number of left cosets of H in G. The left cosets of H in G are in onetoone correspondence with the right cosecs via the
correspondence aH  Ha'' = (aH)l. Therefore, [G : H] is also the number of right cosets of H in G.
(2.17) Theorem. (Lagrange) If H is a subgroup of a finite group G, then [G : H] = (GI/IHI, and in particular, IHI divides IGI. Proof. The left cosets of H partition G into [G : H) sets, each of which has exactly IHI elements. Thus, IGI = [G: H]IHI. (2.18) Corollary. If G is a finite group and a E G then o(a) I IGI. Proof.
1.2 Subgroups and Cosets
13
(2.19) Corollary. If IGI = n, then an = e for all a E G.
0
Proof.
(2.20) Corollary. If IGI = p where p is prime, then G is a cyclic group.
Proof. Choose a E G with a e and consider the subgroup H = (a). Then 0 H 54 {e}, and since IHI I IGI = p, it follows that IHI = p, so H = G.
(2.21) Remark. The converse of Theorem 2.17 is false in the sense that if m is an integer dividing [Cl, then there need not exist a subgroup H of G with IHI = m. A counterexample is given in Exercise 31. It is true, however, when m is prime. This will be proved in Theorem 4.7. (2.22) Definition. If G is any group, then the exponent of G is the smallest
natural number n such that a" = e for all a E G. If no such n exists, we say that G has infinite exponent.
If IGI < oo, then Corollaries 2.18 and 2.19 show that the exponent of G divides the order of G. There is a simple multiplication formula relating indices for a chain of subgroups K C H C G.
(2.23) Proposition. Let G be a group and H, K subgroups with K C H. If [G: K] < oo then
[G: K]= [G:HJ(H:K). Proof. Choose one representative ai (1 < i _< [C : H]) for each left coset of H in G and one representative bj (1 _< j < [H : K]) for each left coset of K in H. Then we claim that the set
{aib3 : l < i < [G : HJ, 1 < j < [H : K]}
consists of exactly one representative from each left coset of K in G. To
see this, let cK be a left coset of K in G. Then c E a1H for a unique ai so that c = aih. Then h E b;K for a unique b; so that c = ajb,k for uniquely determined ai, bj k. Therefore, cK = aibjK for unique ai, b3, and we conclude that the number of left cosets of Kin G is [G: H][H : K]. 0
(2.24) Remark. If IGI < oo then Proposition 2.23 follows immediately from Lagrange's theorem. Indeed, in this case [G : K] = IGI/IKI = (IGI/IHI)(IHI/IKI) = [G: HJ[H : K]. (2.25) Examples. (1)
If G = Z and H = 2Z is the subgroup of even integers, then the cosets of H consist of the even integers and the odd integers. Thus,
Chapter 1. Groups
14
[Z
:
2Z] = 2. Since Z is abelian, it is not necessary to distinguish
between left and right cosets. (2) If G = Z and H = nZ, then [Z : nZ] = n where the coset m+H consists of all integers that have the same remainder as m upon division by n.
(3) Let G = S3 = {e, a, a2, ,0, a,3, 0120) where a = (23 i) and Q = (2 i 3) . If H = (13), then the left cosets of H in G are H = {e, 13}
aH = {a, a,3}
a2H = {a2, a213},
while the right cosets are
H = {e, 0}
Ha = {a, a213}
Ha2 = {a2, a,0}.
Note that, in this example, left cosets are not the same as right cosets.
(4) Let G = GL(2, R) and let H = SL(2, R). Then A, B E GL(2, R) are in the same left coset of H if and only if A'B E H, which means that det(A1B) = 1. This happens if and only if det A = det B. Similarly, A and B are in the same right coset of H if and only if det A = det B. Thus in this example, left cosets of H are also right cosets of H. A set of coset representatives consists of the matrices
j
(5)
a
0J :aER'}
Therefore, the set of cosets of H in G is in onetoone correspondence with the set of nonzero real numbers. Groups of order < 5. Let G be a group with JGJ < 5. If IGI = 1, 2, 3, or 5 then Corollary 2.20 shows that G is cyclic. Suppose now that JGJ = 4. Then every element a 0 e E G has order 2 or 4. If G has an element a of order 4 then G = (a) and G is cyclic. If G does not have any element
of order 4 then G = {e, a, b, c} where a2 = b2 = c2 = e since each nonidentity element must have order 2. Now consider the product ab. If ab = e then ab = a2, so b = a by cancellation. But a and b are distinct
elements. Similarly, ab cannot be a or b, so we must have ab = c. A similar argument shows that ba = c, ac = b = ca, be = a = cb. Thus, G has the Cayley diagram of the Klein 4group. Therefore, we have shown
that there are exactly two nonisomorphic groups of order 4, namely, the cyclic group of order 4 and the Klein 4group. The left cosets of a subgroup were seen (in the proof of Theorem 2.14) to be a partition of G by describing an explicit equivalence relation on G. There are other important equivalence relations that can be defined on a group G. We will conclude this section by describing one such equivalence relation.
(2.26) Definition. Let G be a group and let a, b E G. Then a is conjugate to b if there is a g E G such that b = gag'. It is easy to check that conjugacy
1.3 Normal Subgroups and Isomorphism Theorems
15
is an equivalence relation on G. The equivalence classes are called conjugacy classes. Let [ajc denote the conjugacy class of the element a E G.
(2.27) Proposition. Let G be a group and let a E G. Then I[alci = (G : C(a)]
where C(a) is the centralizer of the element a. Proof. Since gag' = hah' a g'h E C(a)
e:* gC(a) = hC(a),
there is a bijective function 0 : [a]c + G/C(a) = the set of left cosets of 0 C(a), defined by O(gag') = gC(a), which gives the result. (2.28) Corollary. (Class equation) Let G be a finite group. Then
IGJ _ [Z(G)I +> DG : C(a)] where the sum is over a complete set of nonconjugate a not in Z(G). Proof. Since I[a]cl = 1 if and only if a E Z(G), the above equation is nothing more than the partition of G into equivalence classes under conjugation, with the observation that all equivalence classes consisting of a single element have been grouped into [Z(G)I.
1.3 Normal Subgroups, Isomorphism Theorems, and Automorphism Groups If G is a group, let P' (G) denote the set of all nonempty subsets of G and define a multiplication on P'(G) by the formula
S, T E P*(G). Since the multiplication in G is associative it follows that the multiplication in P' (G) is associative, so that parentheses are not necessary in multiplications such as STUV. If S = {s} then we will write sT or Ts instead of {s}T or T{s}. In particular, if H is a subgroup of G and a E G then the left coset aH is just the product in P' (G) of the subsets {a} and H of G and there is no ambiguity in the notation aH. The subset
{e} E P'(G) satisfies eS = Se = S for all S E P'(G). Thus P'(G) has an identity element for its multiplication, namely, {e}, and hence P'(G) forms what is called a monoid (a set with an associative multiplication with an
16
Chapter 1. Groups
identity element), but it is not a group except in the trivial case G = {e} since an inverse will not exist (using the multiplication on P'(G)) for any subset S of G with BSI > 1. If S E PO(G) let S1 = {s'1 : s E S}. Note, however, that S1 is not the inverse of S under the multiplication of P' (G) except when S contains only one element. If H is a subgroup of G, then HH = H, and if IHI < oo, then Remark 2.2 (2) implies that this equality is equivalent to H being a subgroup of G. If H is a subgroup of G then H1 = H since subgroups are closed under inverses. Now consider the following question. Suppose H, K E P' (G) are subgroups of G. Then under what conditions is HK a subgroup of G? The following lemma gives one answer to this question; another answer will be provided later in this section after the concept of normal subgroup has been introduced.
(3.1) Lemma. If H and K are subgroups of G then HK is a subgroup if and
only if HK = KH. Proof. If HK is a subgroup, then HK contains all inverses of elements of
HK. Thus, HK = (HK)'' = K1H1 = KH. Conversely, suppose that HK = KH. Then HK is closed under inverses since (HK)1 = KH = HK, and it is closed under products since
(HK)(HK) = HKHK = HHKK = HK. Thus, HK is a subgroup by Proposition 2.1.
The equality HK = KH is an equality of subsets of G; it should not be confused with element by element commutativity. In terms of elements, HK = KH means that any product hk (h E H, k E K) can also be written k1h1 for some k1 E K, h1 E H. If G is abelian this is of course automatic. We now consider the question of when the subset of P' (G) consisting of all the left cosets of a subgroup H is closed under the multiplication on P. (G).
(3.2) Definition. If H is a subgroup of G then G/H C P'(G) will denote the set of all left cosets of H in G. It is called the coset space of H in G.
Consider two left cosets of H, say aH and bH. If (aH)(bH) = cH, then ab E cH, and hence cH = abH. Therefore, to ask if G/H is closed under multiplication is to ask if the equation (aH)(bH) = abH is true for all a, b E G.
(3.3) Lemma. If H is a subgroup of G, then (aH)(bH) = abH for all a, b E = H for all c E G. G if and only if cHc1
Proof. Suppose cHc 1 = H for all c E G. Then cH = He for all c E G, so
(aH)(bH) = a(Hb)H = a(bH)H = abHH = abH.
1.3 Normal Subgroups and Isomorphism Theorems
17
Conversely, if (aH)(bH) = abH for all a, b E G, then
cHc1 C cHc1H = cc1H = H for all c E G. Replacing c by c1 (since c1 E G) gives an inclusion c 1 He C
H and multiplying on the left by c and the right by c1 gives H C cHc1. Hence, cHc1 = H for all c E G. (3.4) Definition. A subgroup N of G is said to be normal, denoted N a G, if = N for all a E G. aNa1
(3.5) Remark. The argument in Lemma 3.3 shows that N is normal in G if and only if aNa' C N for all a E G. This is frequently easier to check = N. Also note that Definition 3.4 is equivalent than the equality to aN = Na for all a E G. aNa1
(3.6) Proposition. If N a G, then the coset space GIN C P' (G) forms a group under the multiplication inherited from P'(G). Proof. By Lemma 3.3, GIN is closed under the multiplication on P'(G). Since the multiplication on P' (G) is already associative, it is only necessary
to check the existence of an identity and inverses. But the coset N = eN satisfies
(eN)(aN) = eaN = aN = aeN = (aN)(eN), so N is an identity of GIN. Also
(aN)(a1N) = as1N = eN = N = a1aN = (a'N)(aN) so that a1N is an inverse of aN. Therefore, the axioms for a group structure on GIN are satisfied.
(3.7) Definition. If N a G, then GIN is called the quotient group of G by N. (3.8) Remark. If N a G and IGI < oo, then Lagrange's theorem (Theorem 2.17) shows that IG/Nj = [G: N] = IGI/INI. (3.9) Examples. (1) If G is abelian, then every subgroup of G is normal.
(2) SL(n, R) is a normal subgroup of GL(n, R). Indeed, if A E GL(n, R) and B E SL(n, R) then det(ABA1) = (det A)(det B)(det A)1 = 1
Chapter 1. Groups
18
so that ABA1 E SL(n, R) for all A E GL(n, R) and B E SL(n, R). The quotient group GL(n, R)/ SL(n, R) is isomorphic to R', the multiplicative group of nonzero real numbers. This will follow from Theorem 3.11 (to be proved shortly) by considering the homomorphism (3)
(4)
(5)
det : GL(n, R)  R. The details are left as an exercise. The subgroup T(n, R) of upper triangular matrices is not a normal subgroup of GL(n, R). For example, take n = 2 and let A = [ 1 °] and B = [ o 11. Then ABA' = [ °, 2 ] V T(2, R). A similar example can be constructed for any n > 1. Thus the set of cosets GL(n, R)/T(n, R) does not form a group under the operation of coset multiplication. If a = (2 3 ), then H = {e, a, a2} is a normal subgroup of the sym i S3 (check it). If 3 H then the cosets are H and OH. metric group Let K = (f3) C S3 where fi = (2 i 3). Then the left cosecs of K in G are
K = {e. 4}
(Jt : In, ail}
a2K = {a2, a2m
where a is the permutation defined in Example 3.9 (4). Then
K(aK) = {e, a) (a, af31 = {a, af3, a2, a2/} # aK. Therefore, the product of two cosets of K is not a coset of K, and in particular, K is not a normal subgroup of S3. A straightforward calculation shows that aKa1 34 K. (3.10) Proposition. Let f : G a H be a group homomorphism. Then
Ker(f)aG. Proof. Let a E G and b E Ker(f). Then f(aba1) = f(a)f(b)f(a) = f(a)ef(a)1 = e
so aba` 1 E Ker(f) for all b E Ker(f), a E G and Ker(f) is normal by Remark 3.5.
In fact, Proposition 3.10 describes all possible normal subgroups of a group G. To see this let No G and define a function it : G + GIN by the formula 7r(a) = aN. By the definition of multiplication on GIN we see that ir(ab) = abN = (aN)(bN) = 7r(a)a(b).
Thus, n is a group homomorphism (called the natural projection or simply
natural map) from G to GIN. Note that Ker(7r) = N and therefore N is the kernel of a group homomorphism. Since N was an arbitrary normal subgroup of G, it follows that the normal subgroups of G are precisely the kernels of all possible group homomorphisms from G to some other group. We now present some general results, which are commonly called the
noether isomorphism theorems. Similar results will also be seen in the theory of rings and the theory of modules.
1.3 Normal Subgroups and Isomorphism Theorems
19
H be a group (3.11) Theorem. (First isomorphism theorem) Let f : G means is isomorphic Im(f) homomorphism with kernel K. Then G/K to).
Proof. Define a function f : G/K + Im(f) by the formula f (aK) = f (a). The first thing that needs to be checked is that this is a welldefined function
since the coset aK may also be a coset W. It is necessary to check that
f(a) = f(b) in this case. But aK = bK if and only if aIb E K, which means that f (a'b) = e or f (a) = f (b). Therefore, f is a welldefined function on G/K. Also 7((aK)(bK)) = 7(abK) = f (ab) = f (a) f (b) = f (aK) f (bK) so that 7 is a homomorphism. 7 is clearly surjective and Ker(7) = K which is the identity of G/K. Hence f is an isomorphism.
Recall from Lemma 3.1 that the product HK of two subgroups H, K is a subgroup if and only if HK = KH. There is a simple criterion for this commutativity. (3.12) Lemma. Let H, K be subgroups of G. If either H or K is normal in G, then HK is a subgroup of G.
Proof. Suppose K o G. Then aK = Ka for all a e G. In particular, HK = KH, so HK is a subgroup. (3.13) Theorem. (Second isomorphism theorem) Let H and N be subgroups of G with N a G. Then H/(H fl N) HN/N.
Proof. Let r : G + GIN be the natural map and let ro be the restriction of r to H. Then ro is a homomorphism with Ker(ro) = H fl N. Thus,
H/(H fl N) = H/ Ker(ro) = Im(ro). But the image of ro is the set of all cosets of N having representatives in H. Therefore, Im(ro) = HN/N. (3.14) Theorem. (Third isomorphism theorem) Let N 1 since there is an equivalence class J (e, ... , e)). Therefore, there must be other equivalence classes with exactly one element. All of these are of the form {(a, ... , a)} and by the definition of X, such an element of X gives a E G 0 with ap = e.
(4.8) Remark. Note that Corollary 4.6 is a generalization of Proposition 3.16. Proposition 4.3 and its corollaries are useful in beginning a study of the structural theory of finite groups. One use of permutation representations in the structure theory of finite groups is the proof of Cauchy's theorem presented above. The next is in proving the Sylow theorems, which are substantial generalizations of Cauchy's theorem. We begin our presentation of the Sylow theorems by indicating what we mean by an action of a group on a set.
(4.9) Definition. Let G be a group and let X be a set. By an action of G on X we mean a permutation representation 4' : G + SX S. In general, we shall write gx for 4i(g)(x). The fact that 4 is a homomorphism means that
g(hx) = (gh)x for all g, h E G and x E X, while ex = x where e E G is
26
Chapter 1. Groups
the identity. Associated to x E X there is a subset Gx of X and a subgroup G(x) of G defined as follows:
(1) Gx = {gx : g E G} is called the orbit of x. (2) G(x) = {g E G : gx = x} is called the stabilizer of x. (4.10) Lemma. Let the group G act on a finite set X. Then IGxI = [G: G(x)]
for each x E G. Proof. Since
gx=hx t* g1h EG(x) a 9G(x) = hG(x),
there is a bijective function 0 : Gx i GIG(x) defined by 4)(gx) = gG(x), which gives the result.
(4.11) Lemma. Let the group G act on a finite set X. Then
IX I = >[G: G(x)] where the sum is over a set consisting of one representative of each orbit of G.
Proof. The orbits of G form a partition X, and hence IXI = E IGxI where the sum is over a set consisting of one representative of each orbit of G. The result then follows from Lemma 4.10. (4.12) Remark. Note that Lemma 4.11 generalizes the class equation (Corol
lary 2.28), which is the special case of Lemma 4.11 when X = G and G acts on X by conjugation. (4.13) Definition. (1) If p is a prime, a finite group G is a pgroup if IGI = pn for some n > 1.
2) H is a psubgroup of a group G if H is a subgroup of G and H is a pgroup. (3) Let G be an arbitrary finite group, p a prime, and pn the highest power
of p dividing IGI (i.e., pn divides IGI, but pn+l does not). H is a pSylow subgroup of G if H is a subgroup of G and IHI = pn. The three parts of the following theorem are often known as the three Sylow theorems: (4.14) Theorem. (Sylow) Let G be a finite group and let p be a prime dividing IGI.
1.4 Permutation Representations and the Sylow Theorems (1)
(2) (3)
27
G has a pSylow subgroup, and furthermore, every psubgroup of G is contained in some pSylow subgroup. The pSylow subgroups of G are all mutually conjugate. The number of pSylow subgroups of G is congruent to 1 modulo p and divides IGI.
Proof. Let m = IGI and write m = p"k where k is not divisible by p and n > I. We will first prove that G has a pSylow subgroup by induction on M. If m = p then G itself is a pSylow subgroup. Thus, suppose that m > p and consider the class equation of G (Corollary 2.28): (4.1)
Cl I= iZ(G)I + >[G: C(a)]
where the sum is over a complete set of nonconjugate a not in Z(G). There are two possibilities to consider:
(1) For some a, [G : C(a)] is not divisible by p. In that case, IC(a)I = IGI/[G : C(a)] = p"k' for some k' dividing k. Then p divides IC(a)I and IC(a)I < IGI, so by induction C(a) has a subgroup H of order p", which is then also a pSylow subgroup of G. (2) [G: C(a)] is divisible by p for all a Z(G). Then, since IGI is divisible by p, we see from Equation (4.1) that p divides IZ(G)I. By Cauchy's theorem (Theorem 4.7), there is an x E Z(G) of order p. Let N = (x). If n = 1 (i.e., p divides IGI, but p2 does not) then N itself is a pSylow subgroup of G. Otherwise, note that since N C Z(G), it follows that N 4G (Exercise 21). Consider the projection map 7r : G + GIN. Now I G/NI = p"' k < IGI, so by induction, GIN has a subgroup H with IHI = p"'1, and then it1(H) is a pSylow subgroup of G.
Thus, we have established that G has a pSylow subgroup P. Let X be the set of all subgroups of G conjugate to P. (Of course, any subgroup conjugate to P has the same order as P, so it is also a pSylow subgroup of G.) The group G acts on X by conjugation, and since all elements of X are conjugate to P, there is only one orbit. By Lemma 4.11, we have IXI = [G: G(P)]. But P C G(P), so [G: G(P)] divides k and, in particular, is not divisible by p. Thus, IXI is relatively prime to p. Now let H be an arbitrary psubgroup of G, and consider the action of H on X by conjugation. Again by Lemma 4.11, (4.2)
IXI = >[H : H(x)].
Since IXI is not divisible by p, some term on the righthand side of Equation (4.2) must not be divisible by p; since H is a pgroup, that can only happen if it is equal to one. Thus, there is some pSylow subgroup P' of G, conjugate
to P, with hP'h'1 = P' for all h E H, i.e., with HP' = P'H. But then Lemma 3.1 implies that HP is a subgroup of G. Since
Chapter 1. Groups
28
IHP'I = I HII P'I11 H n P'I
(see Exercise 17), it follows that HP is also a psubgroup of G. Since P is a pSylow subgroup, this can only happen if HP = P', i.e., if H C P. Thus part (1) of the theorem is proved. To see that (2) is true, let H itself be any pSylow subgroup of G. Then
H C P for some conjugate P' of P, and since IHI = IP'I, we must have H = P so that H is conjugate to P. This gives that X consists of all the pSylow subgroups of G, and hence, IXI = [G: G(P)] divides IGI. Now take H = P. Equation (4.2) becomes
IXI = E[P : P(x)].
(4.3)
Then, for x = P, [P: P(x)] = 1, while if x is a representative of any other orbit, [P : P(x)] is divisible by p, showing that IXI is congruent to 1 modulo p. Thus part (3) is verified.
The Sylow theorems are a major tool in analyzing the structure of finite groups. In Section 1.7, as an application of these theorems, we will classify all finite groups of order < 15.
1.5 The Symmetric Group and Symmetry Groups Recall that if X = 11, 2, ... , n} then we denote SX by Sn and we can write a typical element a E Sn as a tworowed array
a=
1
2
a(1)
a(2)
it
...
a(n)
This notation is somewhat cumbersome so we introduce a simpler notation which is frequently more useful.
(5.1) Definition. An element i E X = {1, 2, ... , n} is fixed by a E Sn if .x(i) = i. a E Sn is an rcycle or cycle of length r if there are r integers i1,i2i...,ir E X such that
a(il) = i2,
a(i2) = i3,
...
,
a(trt) = ir,
a(ir) = it
and such that a fixes all other i E X. The rcycle a is denoted (il i2 ir). If a is an rcycle, note that o(a) = r. A 2cycle is called a transposition. Two cycles a = (i1 ir) and ,3 = (j1 . j,) are disjoint if {i1,
... ,jr} n {jl, ... ,j8} = 0.
That is, every element moved by a is fixed by ,Q.
1.5 The Symmetric Group and Symmetry Groups
29
As an example of the increased clarity of the cycle notation over the 2rowed notation, consider the following permutation in S9.
_
1
(3
2
9
3 7
4 4
5 2
6 1
7 6
8 8
9 5
a is not a cycle, but it is a product of disjoint cycles, namely, a = (1376)(952)(4)(8). Since 1cycles represent the identity function, it is customary to omit them and write a = (13 7 6) (9 5 2). This expression for a generally gives more
information and is much cleaner than the 2rowed notation. There are, however, two things worth pointing out concerning the cycle notation. First the cycle notation is not unique. For an rcycle (i1 . . jr) there are r different cycle notations for the same rcycle:
(il...ir) = (12i3 ... it E1) = ... = (iril ... it1) The second observation is that the cycle notation does not make it clear which symmetric group S the cycle belongs to. For example, the transposition (12) has the same notation as an element of every Sn for n > 2. In practice, this ambiguity is not a problem. We now prove a factorization theorem for permutations. (5.2) Lemma. Disjoint cycles commute.
Proof. Suppose a and 6 are disjoint cycles in S,,, and let i E X = 11, 2, ... , n}. If i is fixed by both a and Q then of(i) = i = pa(i). If a moves i, then a also moves a(i), and thus, Q fixes both of these elements. Therefore, a0(i) = a(i) = Qa(i). Similarly, if ,Q moves i then a$(i) ='3(i) = /ja(i). (5.3) Theorem. Every a E S with a 34 e can be written uniquely (except for order) as a product of disjoint cycles of length > 2. Proof. We first describe an algorithm for producing the factorization. Let k1 be the smallest integer in X = 11, 2, ... , n} that is not fixed by a (k1 exists since a e) and then choose the smallest positive r1 with a", (k1) = k1 (such an r1 exists since o(a) < oo). Then let a1 be the r1cycle
a1 = (k1 a(k1) a2(k1) ... ari1(k1))
Now let X1 = X \ {k1, a(k1), ... ,ar'1}. If every k E X1 is fixed by a then a = a1 and we are finished. Otherwise
let k2 be the smallest integer in X1 not fixed by a and then let r2 be the smallest positive integer with ar2(k2) = k2. Then let a2 be the r2cycle a2 = (k2 a(k2) a2(k2) ... are1(k2))
Chapter 1. Groups
30
It is clear from the construction that a1 and a2 are disjoint cycles. Continuing in this manner we eventually arrive at a factorization a =
a
into a product of disjoint cycles.
We now consider the question of uniqueness of the factorization. Suppose that
a=a1a2...a.=0102...(jt
where each of these is a factorization of a into disjoint cycles of length
> 2. We must show that s = t and a; = (30lil for some 0 E Sa. Let m = max{s, t}. If m = 1 then a = al = 01 and uniqueness is clear. We proceed by induction on m. Suppose that m > 1 and let k be an element of X that is moved by a. Then some ai and Oj must also move k. Since disjoint cycles commute, we can, without loss of generality, suppose that al and !31 move k. Since none of the other ai or ,0, move k, it follows that
ai(k) = ae(k) = /31(k)
for all f.
Thus, o(al) = o(/31) = r = smallest r with ar(k) = k. Hence,
a1 = (k a(k) ... ar1(k)) = 01. Multiplying by al1 gives a factorization a1
1
a = 02 . . aa = 132 .
. /3e,
and the proof is completed by induction on m.
O
(5.4) Corollary. Every a E S,, is a product of transpositions.
Proof. By Theorem 5.3, it is sufficient to factor any cycle as a product of transpositions. But
(il i2 ... ir) _ (i1 ir)(ii it1) ... (i i2) is such a factorization.
0
In contrast to the uniqueness of the factorization of a permutation into disjoint cycles, writing a permutation as a product of transpositions is not very well behaved. First, the transpositions may not commute. For example,
(13)(12) = (123) 0 (132) = (12)(13). Second, the factorization is not uniquely determined, e.g., (123) = (13)(12) = (13)(12)(23)(23). There is, however, one observation that can be made concerning this factorization;
namely, the number of transpositions occurring in both factorizations is even. While we have shown only one example, this is in fact a result that is true in general. Specifically, the number of transpositions occurring in any factorization of a permutation as a product of transpositions is always odd or always even. This will be verified now.
1.5 The Symmetric Group and Symmetry Groups
31
i,) then a = (il i,.) . (il i2) so that an rcycle a can If a = (il be written as a product of (o(a)  1) transpositions. Hence, if a ,E e E S, a, then a is the product of is written in its cycle decomposition a = al f (a) _ Ei=1(o(a;)  1) transpositions. We also set f (e) = 0. Now suppose that a = (al bl)(a2 b2) ... (at bt)
is written as an arbitrary product of transpositions. We claim that f (a)  t is even. To see this note that
(ai1i2 ... i,.bj ... j,)(ab) _ (a11 ... je)(bil ... ir) and (since (a b)2 = e)
(ail ... js)(bi1 ... i,.)(ab) _ (ail i2 ... i,, b jl ... j,) where it is possible that no ik or jk is present. Hence, if a and b both occur in the same cycle in the cycle decomposition of a it follows that f (a (a b)) = f (a)  1, while if they occur in different cycles or are both not moved by a then f (a (a b)) = f (a) + 1. In any case
f (a (a b))  f (a) = 1
(mod 2).
Continuing this process gives 0 = f (e) = f (a . (atbt) ... (a1b1)) = f (a) + t
(mod 2).
We conclude that any factorization of a into a product of t transpositions has both f (a) and t even or both odd, which is what we wished to verify. Because of this fact we can make the following definition.
(5.5) Definition. A permutation a E Sn is even if a can be written as a product of an even number of transpositions. a is odd if a can be written as a product of an odd number of transpositions. Define the sign of or, denoted sgn(a), by 1 if a is even, sgn(a) 1 if a is odd.
The argument in the previous paragraph shows that a permutation cannot be both even and odd. Thus sgn : Sn  (1, 1) is a welldefined function, and moreover, it is a group homomorphism. The kernel of sgn, i.e., the set of even permutations, is a normal subgroup of Sn called the alternating group and denoted A,,. (5.6) Remark. Note that the above argument gives a method for computing
a, into a product of cycles and compute f(a) = E;=1(o(a;)  1). Then sgn(a) = 1 if f(a) is even and sgn(a) = 1 if f (a) is odd. sgn(a). Namely, decompose a = al
32
Chapter 1. Groups
There is an alternative method that does not require that a be first decomposed into a product of cycles. We have defined a as a bijection of { 1, .. , n}. Let
1(a) = I {(i, j) : 1 < i < j < n and
a(j) < a(i)}I.
Then sgn(a) = 1 if 1(a) is even and sgn(a) = 1 if f (a) is odd. We leave the proof of this as an exercise for the reader. (5.7) Proposition. IAnI = n! /2.
Proof. Since sgn : S  11, 11 is a group homomorphism, the first isomorphism theorem gives Sn/An 95 Im(sgn)
0
Thus, n! = ISnI = 2IAnt.
(5.8) Proposition. If n > 2 then An is generated by all the 3cycles in S.
Proof. An element of A. is a product of terms of the form (ij)(kl) or (i j)(i k) where i, j, k, l are distinct. (If n = 3, only the latter product is possible.) But
(ij)(ik)=(ikj)
and
(ij)(kl) = (ikj)(ikl) so that every element of An is a product of 3cycles.
O
If G is a group recall (Definition 2.26) that two elements a, b E G are conjugate if b = cac 1 for some c E G. In general, it is not easy to determine if two elements of G are conjugate, but for the group Sn there is a simple criterion for conjugacy based on the cycle decomposition (factorization) of
a, Q E S. We will say that a and Q have the same cycle structure if their factorizations into disjoint cycles produce the same number of rcycles for each r.
(5.9) Proposition. (1) If a E Sn and a = (il a$a'1 is the rcycle (a(ii) ... a(ir)). (2) Any two rcycles in Sn are conjugate.
ir) is an rcycle, then
Proof. (1) If j 4 (a(il), ... ,a(ir)) then a1(j) is fixed by Q so that a/a1(j) = j. Also aoal(a(il)) = a(i2) aQa1(a(ir1)) = a(ir)
aAa'(a(ir)) =a(il) a/a1 so that = (a(il) ... a(ir)).
1.5 The Symmetric Group and Symmetry Groups
33
i,.) and y = (jl . j,.) be any two rcycles in Sn. (2) Let Q = (i1 Define a E S,, by a(ik) =.1k for 1 < k < r and extend a to a permutation apa1 = ry. in any manner. Then by part (1) (5.10) Corollary. Two permutations a,,Q E Sn are conjugate if and only if they have the same cycle structure. a, is the cycle decomProof. Suppose that ryary1 = ,3. Then if a = a1 position of a, it follows from Proposition 5.9 (1) that Q ='ya71 = (7a171)(ya271) ... ('Ya8i ') is the cycle decomposition of Q. Thus, a and 0 have the same cycle structure. The converse is analogous to the proof of Proposition 5.9 (2); it is left to the reader.
(5.11) Example. Let H = {e, (12)(34), (13)(24), (14)(23)} C S4. Then H is a subgroup of S4 isomorphic with the Klein 4group, and since H consists of all permutations in S4 with cycle type (a b) (c d) (where a, b, c, d are all distinct), it follows from Corollary 5.10 that HaS4. Let K = {e, (12)(34)}. Then K is a normal subgroup of H (since H is abelian), but K is not normal
in S4 since any other permutation of cycle type (ab)(cd) can be obtained from (12)(34) by conjugation in S4. Therefore, normality is not a transitive property on the set of all subgroups of a group G.
Let X C R". By a symmetry of X we mean a function f : R" + R" such that f (X) C X and IIx  yfl = IIf (x)  f (y)II for all x, y E R". The set of all symmetries of X under functional composition forms a group, called the symmetry group of X. If X = P" C R2 is a regular polygon with n vertices then a symmetry is completely determined by the action on the vertices (since it is easy to see from the triangle inequality that lines must go to lines and adjacent vertices must go to adjacent vertices) so that we get a permutation representation of the symmetry group of P", denoted Den, as a subgroup of S,i. D2n is called the dihedral group of order 2n. If P,, is taken on the unit circle centered at (0, 0) with one vertex
at (1,0) then the symmetries of Pn are the rotations through an angle of Ok = 2k7r/n around (0, 0) for 0 < k < n and the reflections through the lines from each vertex and from the midpoint of each side to the center of the circle. (There are always n such distinct lines.) Thus ID2nI = 2n when there are n rotations and n reflections. If we let a be the rotation through the angle 01 and Q the reflection through the xaxis, then
D2n={a'( :0 q and let G be a group of order pq. (1) If q does not divide p  1, then G Z. (2) If q I p  1, then G Zpq or G Zp x m
Z. when
0: Zq ' Aut(Zp) = Z. is a nontrivial homomorphism. All nontrivial homomorphisms produce isomorphic groups.
Proof. By Cauchy's theorem (Theorem 4.7) G has a subgroup N of order p and a subgroup H of order q, both of which are necessarily cyclic. Then N aG since (G : N) = q and q is the smallest prime dividing IGI (Corollary 4.6). Since it is clear that N n H = (e) and NH = G, it follows that G is the semidirect product of N and H. The map 0 : H  Aut(N) given by Oh (n) = hnh is a group homomorphism, so if q does not divide I Aut(N)I = I Aut(Zp)l = 1Z;1 = p  1,
then 0 is the trivial homomorphism. Hence Oh = IN for all h E H, i.e., nh = hn for all h E H, n E N. Hence H a G and G Zp x Z. ?! Zpq (see Exercise 11). If q I p  1 then there are nontrivial homomorphisms
0: Zq  Aut(N)
Zp
and for some homomorphism 0, G
ZP 14m Z.
Therefore, if N = (a) and H = (b), then G = (a, b), subject to the relations ap = e,
bq = e,
blab = ar
where rq  1 (mod p). If r = 1 then 0 is trivial, H is normal, and G ZP X Z. Otherwise, G is nonabelian. We leave it as an exercise to verify
1.7 Groups of Low Order
41
that all choices of r 54 1 produce isomorphic groups. Thus, if q I p  1, then there are exactly two nonisomorphic groups of order pq.
(7.3) Corollary. If IGI = 2p, where p is an odd prime, then G G
Z2p or
D2p.
Proof. The only nontrivial homomorphism 0: Z2 i Aut(Zp) = Zp is the homomorphism 1 * 01 with ¢1(a) = a. Apply Example 6.10 (2). (7.4) Remark. The results obtained so far completely describe all groups of order < 15, except for groups of order 8 and 12. We shall analyze each of these two cases separately.
Groups of Order 8 We will consider first the case of abelian groups of order 8.
(7.5) Proposition. If G is an abelian group of order 8, then G is isomorphic to exactly one of the following groups: (1) Z8,
(2) Z4 X Z2, or (3) Z2 X Z2 X Z2.
Proof. Case 1: Suppose that G has an element of order 8. Then G is cyclic and, hence, isomorphic to Z8. Case 2: Suppose every element of G has order 2. Let {a, b, c} C G\ {e} with c j4 ab. Then H = (a, b) is a subgroup of G isomorphic to Z2 X Z2. Furthermore, H n (c) _ (e) and H(c) = G so that
GL, Hx (c)S: Z2xZ2xZ2. Case 3: If G does not come under Case 1 or Case 2, then G is not cyclic and not every element has order 2. Therefore, G has an element a of order 4. We claim that there is an element b V (a) such that b2 = e. To see this, let c be any element not in (a). If c2 = e, take b = c. Otherwise, we must have o(c) = 4. Since IG/(a)l = 2, it follows that c2 E (a). Since a2 is the only element of (a) of order 2, it follows that c2 = a2. Let b = ac. Then b2 = a2c2 = a4 = e.
Proposition 6.3 then shows that
C
Z4xZ2.
Since every abelian group of order 8 is covered by Case 1, Case 2, or Case 3, the proof is complete.
42
Chapter 1. Groups
Now consider the case of nonabelian groups of order 8.
(7.6) Proposition. If G is a nonabelian group of order 8, then G is isomorphic to exactly one of the following two groups:
(1) Q = the quaternion group, or (2) D8 = the dihedral group of order 8. Proof. Since G is not abelian, it is not cyclic so G does not have an element of order 8. Similarly, if a2 = e for all a E G, then G is abelian (Exercise 8); therefore, there is an element a E G of order 4. Let b be an element of G not in (a). Since (G : (a)] = 2, the subgroup (a) a G. But IG/(a)I = 2 so that b2 E (a). Since o(b) is 2 or 4, we must have b2 = e or b2 = a2. Since (a) a C, blab is in (a) and has order 4. Since G is not abelian, it follows that blab = a3. Therefore, G has two generators a and b subject to one of the following sets of relations: (1) a4 = e,
b2 = e,
(2) a4 = e,
b2 = a2,
blab = a3;
blab = a3.
In case (1), G is isomorphic to Ds, while in case (2) G is isomorphic to Q. We leave it as an exercise to check that Q and D8 are not isomorphic.
0 (7.7) Remarks. (1) Propositions 7.5 and 7.6 together show that there are precisely 5 distinct isomorphism classes of groups of order 8; 3 are abelian and 2 are nonabelian. (2) D8 is a semidirect product of Z4 and Z2 as was observed in Example 6.10 (2). However, Q is a nonsplit extension of Z4 by Z2, or of Z2 by Z2 X Z2.
In fact Q is not a semidirect product of proper subgroups.
Groups of Order 12 To classify groups of order 12, we start with the following result.
(7.8) Proposition. Let G be a group of order p2q where p and q are distinct primes. Then G is the semidirect product of a pSylow subgroup H and a qSylow subgroup K.
Proof. If p > q then H a G by Corollary 4.6.
If q > p then 1 + kq p2 for some k > 0. Since q > p, this can only occur if k = 0 or 1 + kq = p2. The latter case forces q to divide p21 = (p+l)(p1). Since q > p, we must have q = p+l. This can happen I
only if p _ 2 and q = 3. Therefore, in the case q > p, the qSylow subgroup K is a normal subgroup of G, except possibly when JGI = 22  3 = 12.
1.7 Groups of Low Order
43
To analyze this case, let K be a 3Sylow subgroup of a group G of order 12. If K is not normal in G, then the number of 3Sylow subgroups of G is 4. Let these 3Sylow subgroups be K1, K2, K3, and K4. Then K1 U K2 U K3 U K4 accounts for 9 distinct elements of G.
The remaining elements, together with the identity e, must form the 2Sylow subgroup H of G. Hence, we must have H< G. Therefore, we have shown that at least one of H (a pSylow subgroup of G) or K (a qSylow subgroup of G) is normal in G. Since it is clear that H fl K = (e) and HK = G, it follows that G is a semidirect product of H
and K. (7.9) Proposition. A nonabelian group G of order 12 is isomorphic to exactly one of the following groups: (1) A4,
(2) D12, or
(3) T = Z3 AS Z4 where
:
Z4
Aut(Z3)
Z2 is the nontrivial
homomorphism.
Proof. Let H be a 2Sylow subgroup and K a 3Sylow subgroup of G. By Proposition 7.8 and the fact that G is nonabelian, exactly one of H and K is normal in G. Case 1: Suppose H 4 G. Then K is not normal in G. Since [G : K] = 4, G  S4. By Proposition 4.3, there is a permutation representation Ker(IiK) is the largest normal subgroup of G contained in K. Since K has prime order and is not normal, it follows that G is injective so that IM(4)K) C S4.
G
It is an easy exercise to show that A4 is the only subgroup of S4 of order 12; therefore, G A4 if the 2Sylow subgroup is normal in G. Case 2: Suppose K< G and H Z4. In this case G
Z3 is 0 Z4
where 0: Z4 Aut(K) is a nontrivial homomorphism, but the only nontrivial automorphism of Z3 is a ' a1 where K = (a). In this case G T. Case 3: Suppose K,* G and H °_5 Z2 X Z2. Let K = (a) and let
0:H.Aut(K)2 Z2 be the conjugation homomorphism. Then H = (Ker(O)) X Z2, so let
Ker(0) = (c) and let d E H with 0(d) 34 1K. Then clac = a and dlad = a1 = a2. Let b = ac. Then o(b) = 6, d V (b), and
d'bd = dlacd = dlade = a2c = (ac)1 = b1. Thus, G
D12.
44
Chapter 1. Groups
It remains to consider the case of abelian groups of order 12.
(7.10) Proposition. If G is an abelian group of order 12, then G is isomorphic to exactly one of the following groups: (1) Z12, or
(2) Z2 X Z6. Proof. Exercise.
By combining the results of this section we arrive at the following table
of distinct groups of order at most 15. That is, every group of order < 15 is isomorphic to exactly one group in this table.
Table 7.1. Groups of order < 15
Order
Abelian Groups
Nonabelian Groups
Total Number
I
{e}
1
2
Z2 Z3 Z4
1
3 4
1
2
Z2xZ2 5
6 7
8
Z5 Z6 Z7 Z8
Z4 X Z2
1
S3
2
Q
5
1
D8
Z2 1, since 1 + 1j = R, we can find bj E 11 and cj E Ij with bj + Cj = 1. Then fl 2(b; + cj) = 1, and since each b j E It, it follows that 1 = [J .2(bi + cj) E 11 + [I 2 I. Therefore, there
Chapter 2. Rings
66
is a1 E I1 and /31 E 11 =2 Ii such that al + 01 = 1. Observe that j31 solves the required congruences in the special case under consideration. That is,
(311 (mod I,)
/310 (mod1j)
for
j
1
since /31  1 E I1 and /31 E Ijj=2 Ii C II for i 0 1. By a similar construction we are able to find /32,. .. , /3n such that ,Ui = 1
Qi  0 Now let a = a1131 +
(mod Ii) (mod II) for j # i.
+ a
ai (modli) (1 0 > co, and thus, f (X)g(X) A 0.
(2) If f (X)g(X) = 1 then deg(f (X)) + deg(g(X)) = deg(1) = 0. Thus, f (X) and g(X) are both polynomials of degree 0, i.e., elements of R. Therefore, they are units not only in R[X] but in R also.
We now consider the division algorithm for R[X] where R is a commutative ring. Let f (X) = ao + a1X + + anX" E R[X] and let g(X) = b0+b1X + +bm_1Xm_1 +Xm be a monic polynomial in R[X] of degree
m > 1. If n > m, let q1(X) = anX"m. Then f1(X) = f(X)  g(X)gl(X) has degree < n  1. If deg(fl (X)) > m then repeat the process with f (X) replaced by f 1(X ). After a finite number of steps we will arrive at a poly
nomial f. (X) of degree < m. Letting q(X) = ql (X) + + q,(X) and r(X) = f (X)  g(X)q(X) we obtain an equation f (X) = g(X)q(X) + r(X) where deg(r(X)) < deg(g(X)). What we have described is the familiar long division process for polynomials.
(4.4) Theorem. (Division algorithm) Let R be a commutative ring, let f (X) E R[X ], and let g(X) E R[X J be a monic polynomial. Then there are
unique polynomials q(X) and r(X) in R[X] with deg(r(X)) < deg(g(X)) such that f (X) = g(X)q(X) + r(X). Proof. Existence follows from the algorithm described in the previous paragraph. Now consider uniqueness. Suppose there are two such decomposi
tions f(X) = g(X)q(X) + r(X) and f(X) = g(X)gl(X) + rl(X) with deg(r(X)) < deg(g(X)) and deg(rl(X)) < deg(g(X)). Then
2.4 Polynomial Rings
75
g(X)(gi(X)  q(X)) = r(X)  rl(X) Since g(X) is a monic polynomial, taking degrees of both sides gives
deg(g(X)) +deg(gl(X)  q(X)) = deg(r(X)  rl(X)) < deg(g(X)). This forces deg(gl (X)  q(X)) = oo, i.e., ql (X) = q(X). It then follows that rl (X) = r(X), so uniqueness is established. (4.5) Corollary. (Remainder theorem) Let R be a commutative ring and let a E R. Then for any f (X) E R[X]
f (X) = (X  a)q(X) + f (a) for some q(X) E R[X]
Proof. By the division algorithm we may write f (X) = (X  a)q(X) + r(X)
where deg(r(X)) < 0. Therefore, r(X) = r E R. Apply the substitution homomorphism X a to get f (a) = (a  a)q(a) + r so that r = f (a). (4.6) Corollary. Let R be a commutative ring, f (X) E R[X], and let a E R. Then f (a) = 0 if and only if X  a divides f (X). Proof.
(4.7) Corollary. Let R be an integral domain and let f (X) # 0 E R(X] be a polynomial of degree n. Then there are at most n roots of f (X) in R, i.e., there are at most n elements a E R with f (a) = 0.
Proof. If n = 0 the result is certainly true since f (X) = ao 54 0 and thus f (a) = ao 54 0 for every a E R, i.e., f (X) has no roots. Now let n > 0 and suppose, by induction, that the result is true for polynomials of degree < n. If there are no roots of f (X) in R, then there is nothing to prove. Thus suppose that there is at least 1 root a E R. Then by Corollary 4.6, X  a divides f (X), so we may write f (X) = (X  a)q(X) where deg(q(X)) = n  1. By the induction hypothesis, there are at most n  1 roots of q(X) in R. Now let b be any root of f (X) so that 0 = f (b) = (b  a)q(a). Hence b = a or b is a root of q(X). We conclude that f (X) has at most (n  1) + 1 = n roots in R. (4.8) Remark. This result is false if R is not an integral domain. For example
if R = Z2 X Z2 then all four elements of R are roots of the quadratic polynomial X2  X E R[X].
(4.9) Corollary. Let R be an integral domain and let f (X), g(X) E R[X] be polynomials of degree  < n. If f (a) = g(a) for n + 1 distinct a E R, then
f(X) =g(X)
Chapter 2. Rings
76
Proof. The polynomial h(X) = f (X)  g(X) is of degree < n and has 0
greater than n roots. Thus h(X) = 0 by Corollary 4.7.
In the case R is a field, there is the following complement to Corollary 4.9.
(4.10) Proposition. (Lagrange Interpolation) Let F be afield and let ao, a1, ..., an be n + 1 distinct elements of F. Let co, c1, ..., Cn be arbitrary (not necessarily distinct) elements of F. Then there exists a unique polynomial f (X) E F[X] of degree < n such that f (a,) = ci for 0 < i < n.
Proof. Uniqueness follows from Corollary 4.9, so it is only necessary to demonstrate existence. To see existence, for 0 < i < n, let P, (X) E FIX] be defined by
Pi(X) _ f (X  ai)
(4.1)
(aj  aj)
Note that deg Pi (X) = n and P, (ai) = b,i. Thus, we may take n
f(X) _ >c:Pi(X)
(4.2)
i=o
0
to conclude the proof.
(4.11) Remark. The polynomial given in Equation (4.2) is known as Lagrange's form of the interpolation polynomial. Of course, the interpolation polynomial is unique, but there is more than one way to express the interpolation polynomial; in the language of vector spaces (see Chapter 3), this is simply the observation that there is more than one choice of a basis of the vector space Pn(F) of polynomials in F[X] of degree at most n. The set {Pi(X) : 0 n then a E I, = In so that a = clam + 
E I in case r < n. '

Then, as above, deg
(1(x)
 C"1 CiXrnfniX ) i=1
and hence f (X) E I by induction on r. Thus, I C I and the other inclusion is clear. Hence I is finitely generated. 0 (4.15) Corollary. Let R be a commutative ring in which every ideal is finitely generated. Then every ideal of R[X1, ... , Xn] is finitely generated.
Proof. This follows from Theorem 4.14 by induction on it since it is an easy exercise to verify that
R[X1, ... , Xn]
(R[X 1, ... , Xn1]) [X.]
0 (4.16) Corollary. Let F be a field. Then every ideal in the polynomial ring F[X1,
... , Xn]
is finitely generated. Proof
0
2.5 Principal Ideal Domains and Euclidean Domains
79
The theory of principal ideal domains will be studied in Section 2.5. At the present time we have two major examples of PIDs, namely, Z and F[X) for F a field, to which the theory will apply. Further examples will be given in Section 2.5. We will conclude this section with the following concept, which is defined by means of polynomials.
(4.17) Definition. Let F be a field. F is said to be algebraically closed if every nonconstant polynomial f (X) E FIX) has a root in F. (4.18) Remarks.
(1) According to Corollary 4.6, F is algebraically closed if and only if the nonconstant irreducible polynomials in FIX) are precisely the polynomials of degree 1. (2) The complex numbers C are algebraically closed. This fact, known
as the fundamental theorem of algebra, will be assumed known at a number of points in the text. We shall not, however, present a proof. (3) If F is an arbitrary field, then there is a field K such that F C K and
K is algebraically closed. One can even guarantee that every element
of K is algebraic over F, i.e., if a E K then there is a polynomial f (X) E FIX) such that f (a) = 0. Again, this is a fact that we shall not prove since it involves a few subtleties of set theory that we do not wish to address.
2.5 Principal Ideal Domains and Euclidean Domains The fundamental theorem of arithmetic concerns the factorization of an integer into a unique product of prime numbers. In this section we will show that the fundmental theorem of arithmetic is also valid in an arbitrary principal ideal domain. At present we have only two major examples of PIDs, namely, Z and F[X] for F a field. Some examples will be presented of other PIDs. We will start by defining precisely the concepts of factorization
needed to state and prove the extension of the fundamental theorem of arithmetic.
(5.1) Definition. Let R be an integral domain and let a, b E R \ {0).
(1) a and b are associates if a = ub for some unit u E R. We can define an equivalence relation on R by setting a  b if and only if a and b are associates in R. (2) a divides b (written a I b) if b = ac for some c E R.
(3) A nonunit a is irreducible if a = be implies that b or c is a unit. (4) A nonunit a is prime if a be implies that a I b or a c.
80
Chapter 2. Rings
(5.2) Remark. Let R be an integral domain and let a, b E R \ {0}. (1) a and b are associates if and only if a I b and b I a. (2) Recall that (a) denotes the ideal of R generated by a. Then a I b if and only if (b) C (a) and a and b are associates if and only if (a) = (b). (3) a is a prime element of R if and only if (a) is a prime ideal. (4) If a I b then au by for any units u, v. (5) If a l b and a is not a unit, then b is not a unit. Indeed, if b is a unit then be = 1 for some c E R and b = ad since a I b. Then (ad)c = 1 = a(dc) so, a is a unit also. an then p I ai for some i (exercise). (6) If p is a prime in R and p ( al
(5.3) Lemma. Let R be an integral domain. If a E R is prime, then a is irreducible.
Proof. Let a E R be prime and suppose that a = be. Then a I b or a I c. To be specific, suppose that a I b so that b = ad for some d E R. Then a = be = adc, and since R is an integral domain, Lemma 1.3 shows that 0 dc = 1 so that c is a unit. Thus a is irreducible. If R = Z then the concepts of prime and irreducible are the same, so that the converse of Lemma 5.3 is also valid. In fact, we shall show that the converse is valid in any PID, but it is not valid for every integral domain as the following examples show. (5.4) Examples. (1) Let F be a field and let
R = F[X2, X3J = {p(X) E F[XJ : p(X) = as + a2X 2 + a3 X3 +
+ anX"}.
Then X2 and X3 are irreducible in R, but they are not prime since X2 I (X3)2 = X6, but X2 does not divide X3, and X3 I X4X2 = X6, but X3 does not divide either X4 or X2. All of these statements are easily seen by comparing degrees. (2)
Let Z[/1 = {a + b/ :
a, b E Z}. In Z[f] the element 2 is
irreducible but not prime.
Proof. Suppose that 2 = (a + bs)(c + dy) with a, b, c, d E Z. Then taking complex conjugates gives 2 = (a  bs)(c  d%), so multiplying these two equations gives 4 = (a2 + 3b2)(c2 + 3d2). Since the equation in integers a2 + 302 = 2 has no solutions, it follows that we must have a2 +3b2 = 1 or c2 +3d2 = 1 and this forces a = ±1, b = 0 or c = ±1, d = 0. Note that 2 is not a unit in Z[v/3] Therefore, 2 is irreducible in Z[
since the equation 2(a + b/) = 1 has no solution with a, b E Z.
2.5 Principal Ideal Domains and Euclidean Domains
81
To see that 2 is not prime, note that 2 divides 4 = (1 + V" 3)(1 ../3)
but 2 does not divide either of the factors 1 + vJ or 1 We conclude that 2 is not a prime in the ring Z[3J.
in Z[/).
D
(5.5) Definition. Let R be an integral domain and let A be a subset of R containing at least one nonzero element. We say that d E R is a greatest common divisor (gcd) of A if (1) d l a for all a E A, and (2)
if eERand eI a forallaEA, then eI d.
If 1 is a gcd of A, then we say that the set A is relatively prime. We say that m 54 0 E R is a least common multiple (1cm) of A if 0 V A, if
(1) al m forallaEA, and (2)
if eE R and aIe for all aE R, then mIe.
Note that any two gcds of A are associates. Thus the gcd (if it exists) is well defined up to multiplication by a unit. The following result shows that in a PID there exists a gcd of any nonempty subset of nonzero elements.
(5.6) Theorem. Let R be a PID and let A be a subset of R containing at least one nonzero element.
(1) An element d E R is a gcd of A if and only if d is a generator for the ideal (A) generated by A. ... , an} is finite and a; 0 for 1 < i < n, then an element m E R is a lcm of A if and only if m is a generator of the ideal
(2) If A = {al,
(al) n ... (1 (an),
Proof. (1) Suppose that d is a generator of the ideal (A). Certainly d I a for each a E A since a E (A) = (d). Also, since d E (A), it follows that d = E , riai for rl,... , rn E R and al, ... , an E A. Therefore, if e I a for
allaE A, thene I dsothatdisagcdofA. Conversely, suppose that d is a gcd of the set A and let (A) = (c). Then d I a for all a E A so that a E (d). Hence, (c) = (A) C (d).
But, for each a E A, a E (c) so that c I a. Since d is a gcd of A, it follows that c I d, i.e., (d) C (c). Hence (c) = (d) and d is a generator of the ideal (A).
(2) Exercise.
0
(5.7) Corollary. Let R be a PID and a E R \ {0}. Then a is prime if and only if a is irreducible.
82
Chapter 2. Rings
Proof. Lemma 5.3 shows that if a is prime, then a is irreducible. Now assume that a is irreducible and suppose that a I be. Let d = gcd{a, b}. Thus a = de
and b = df . Since a = de and a is irreducible, either d is a unit or e is a unit. If e is a unit, then a I b because a is an associate of d and d I b. If d is a unit, then d = ar' + bs' for some r', s' E R (since d E (a, b)). Therefore,
I = ar + bs for some r, 8 E R, and hence, c = arc + bsc. But a I arc and a ( bsc (since a I be by assumption), so a I c as required.
(5.8) Corollary. Let R be a PID and let I C R be a nonzero ideal. Then I is a prime ideal if and only if I is a maximal ideal. Proof. By Theorem 2.21, if I is maximal then 1 is prime. Conversely, suppose that I is prime. Then since R is a PID we have that I = (p) where p is a prime element of R. If I C J = (a) then p = ra for some r E R. But p is prime and hence irreducible, so either r or a is a unit. If r is a unit then
(p) = (a), i.e., I = J. If a is a unit then J = R. Thus 1 is maximal. (5.9) Definition. Let R be a ring. We say that R satisfies the ascending chain condition (ACC) on ideals if for any chain
Il 9129139... of ideals of R there is an n such that Ik = In for all k > n, i.e., the chain is eventually constant. A ring that satisfies the ascending chain condition is said to be Noetherian. The following characterization of Noetherian rings uses the concept of maximal element in a partially ordered set. Recall what this means (see the appendix). If X is a partially ordered set (e.g., X C P(Y) where the partial order is set inclusion), then a maximal element of X is an element m E X such that if m < x for some x E X, then m = x. That is, a element m E X is maximal if there is no element strictly larger than m in the partial order of X. For example, if X = {(2), (3), (12) } is a set consisting of the given ideals of the ring Z with the partial order being inclusion of sets, then both (2) and (3) are maximal elements of X.
(5.10) Proposition. Let R be a ring. The following conditions on R are equivalent.
(1) R is Noetherian. (2) Every ideal of R is finitely generated. (3) Every nonempty subset of ideals of R has a maximal element. In particular, a PID is Noetherian. Proof. (1) . (3) Suppose that S = {IQ}QEA is a nonempty set of ideals of R that does not have a maximal element.
2.5 Principal Ideal Domains and Euclidean Domains
83
Then choose 11 E S. Since S does not have a maximal element, there is an element 12 E S such that 11 5 12. Similarly, 12 is not a maximal element
so there is 13 E S such that 12 5 13. In this manner we can construct a strictly increasing chain of ideal in R, which contradicts the assumption that R satisfies the ACC. Therefore, S must contain a maximal element if R satisfies the ACC.
(3) = (2) Let I be an ideal of R and consider the family S of all finitely generated ideals of R that are contained in I. By hypothesis, there is a maximal element J E S. Let a E I. Then the ideal J + (a) E S and it contains J. Since J is maximal in S, it follows that J = J + (a), i.e., a E J. Therefore, I = J and I is finitely generated. (2) * (1) Suppose that every ideal of R is finitely generated. Let
119129139 ... be a chain of ideals of R and let I = U°_1 I. Then I is an ideal of R so that I = (a1i ... , am) for some ai E R. But ai E I = U°O_1 In for 1 < i < m so ai E In, for some ni. Since we have a chain of ideals, it follows that there is an n such that ai E In for all i. Thus, for any k > n there is an inclusion
(al, ... , am) C Ik C 1 = (al, ... , am) so that Ik = I = In for all k > n and R satisfies the ACC on ideals.
0
(5.11) Remark. In light of Definition 5.9 and Proposition 5.10, the Hilbert basis theorem (Theorem 4.14) and its corollary (Corollary 4.15) are often stated:
If R is a commutative Noetherian ring, then the polynomial ring RIX1i
... ,XnJ is also a commutative Noetherian ring.
(5.12) Theorem. (Fundamental theorem of arithmetic) Let R be a PID. Then any nonzero a E R can be written as a = up1 . . . pn where u is a unit and each pi is a prime. Moreover, this factorization is essentially unique. That is, if a = vQ1 . . . qm where v is a unit and each qi is a prime, then m = n and for some permutation a E Sn, qi is an associate of po(;) for
1 0 and suppose that the result is true for all elements b E R that have a factorization with fewer than k prime elements. Then qm, so pn divides some qj since pn is a prime element. After reordering, if necessary, we can assume that pn I qm. But q,n is prime, so qm = Pnc implies that c is a unit. Thus, pn and qn are associates. Let Pn I go91
a'
Then a' has a factorization with fewer than k prime factors, so the induction
hypothesis implies that n  1 = m  1 and qi is an associate of p,(;) for some a E Sn _ 1 i and the argument is complete.
(5.13) Corollary. Let R be a PID and let a 0 0 E R. Then
a=upli ...pkk where P' .. , pA, are distinct primes and u is a unit. The factorization is essentially unique. Proof.
(5.14) Remark. Note that the proof of Theorem 5.12 actually shows that if R is any commutative Noetherian ring, then every nonzero element a E R has a factorization into irreducible elements, i.e., any a E R can be factored as a = up1 ... pn where u is a unit of R and p1, ... , pn are irreducible (not prime) elements, but this factorization is not necessarily unique; however, this is not a particularly useful result.
2.5 Principal Ideal Domains and Euclidean Domains
85
For nonNoetherian rings we do not even have the factorization into irreducible elements. Examples 5.15 (4) and (5) are examples. (5.15) Examples. (1)
(2)
In F[X2, X3] there are two different factorizations of X6 into irreducible elements, namely, (X2)3 = Xs = (X3)2.
In Z[v there is a factorization
4=2.2=(1+v)(1v/'3) into two essentially different products of irreducibles. (3)
If F is a field and p(X) E F[X) is an irreducible polynomial, then p(X) is prime since F[X] is a PID. Thus the ideal (p(X)) is maximal according to Corollary 5.8. Hence the quotient ring F[X]/(p(X)) is a field. If p E Z is a prime number let Fp denote the field Zp. Then F2[X)/(X2+X+1) is a field with 4 elements, F3[X]/(X2+1) is a field with 9 elements, while F2[X]/(X3 + X + 1) is a field with 8 elements. In fact, one can construct for any prime p E Z and n > 1 a field F. with q = p" elements by producing an irreducible polynomial of degree
n in the polynomial ring Fp[X]. (It turns out that F. is unique up to isomorphism, but the proof of this requires Galois theory, which we do
not treat here.) (4) Let H(C) be the ring of complex analytic functions on the entire complex plane. (Consult any textbook on complex analysis for verification of the basic properties of the ring H(C).) The units of H(C) are precisely the complex analytic functions f : C C such that f (z) 0 0
for all z E C. Furthermore, if a E C then the function z  a divides f (z) E H(C) if and only if f (a) = 0. From this it is easy to see (exercise) that the irreducible elements of H(C) are precisely the functions (z  a) f (z) where f (z) 34 0 for all z E C. Thus, a complex analytic function g(z) can be written as a finite product of irreducible elements if and only if g has only finitely many zeros. Therefore, the complex analytic function sin(z) cannot be written as a finite product of irreducible elements in the ring H(C). (Incidentally, according to Remark 5.14, this shows that the ring H(C) is not Noetherian.) (5) Let R be the subring of Q[X] consisting of all polynomials whose constant term is an integer, i.e.,
R are the constant polynomials ±1. Note that for any nonzero integer k, there is a factorization X = k(X/k) E R, and neither
factor is a unit of R. This readily implies that X does not have a factorization into irreducibles. (Again, Remark 5.14 implies that R is not Noetherian, but this is easy to see directly:
86
Chapter 2. Rings
...
(X) 5 (X/2) 5 (X/4) 5 (X/8) is an infinite ascending chain.)
Both of the examples of PIDs that we currently have available, namely,
Z and FIX] for F a field, actually have more structure than just that of a PID. Specifically, both Z and F(X] have a measure of the size of elements, Inl for n E Z and deg(p(X)) for p(X) E FIX], together with a division algorithm, which allows one to divide one element by another with a remainder "smaller" than the divisor. In fact the division algorithm was
precisely what was needed to prove that Z and F(X] are principal ideal domains. We formalize this property of Z and FIX] with the following definition.
(5.16) Definition. An integral domain R is a Euclidean domain if there is a Z+ = N U {0} such that function v : R \ {0} (1) for all a, b E R \ {0}, v(a) < v(ab); and (2) given a, b E R with a # 0, there are q, r E R with b = aq + r such that
r = 0 or v(r) < v(a). (5.17) Examples.
(1) Z together with v(n) = Ink is a Euclidean domain. (2)
If F is a field, then FIX] together with v(p(X)) = deg(p(X)) is a Euclidean domain (Theorem 4.4).
(5.18) Lemma. If R is a Euclidean domain and a E R\{0}, then v(1) < v(a). Furthermore, v(1) = v(a) if and only if a is a unit.
Proof. First note that v(1) < v(1a) = v(a). If a is a unit, let ab = 1. Then v(a) < v(ab) = v(1), so v(1) = v(a). Conversely, suppose that v(1) = v(a) and divide 1 by a. Thus, 1 = aq+r where r = 0 or v(r) < v(a) = v(1). But v(1) < v(r) for any r 0 0, so the latter possibility cannot occur. Therefore,
0
r= 0 so l= aq and a is a unit. (5.19) Theorem. Let R be a Euclidean domain. Then R is a PID.
Proof. Let I C R be a nonzero ideal and let S={v(a):aEI\{0}}CZ+.
This set has a smallest element no. Choose a E I with v(a) = no. We claim
that I = Ra. Since a E I, it is certainly true that Ra C I. Now let b E I.
Then b=aq+rforq,rE R with r = 0 or v(r) < v(a). Butr=b  aqE I, so v(a) < v(r) if r 0. Therefore, we must have r = 0 so that b = aq E Ra. 0 Hence, I = Ra is principal.
2.5 Principal Ideal Domains and Euclidean Domains
87
In a Euclidean domain R the classical Euclidean algorithm for finding the gcd of two integers works for finding the gcd of two elemen a of R. This algorithm is the following. Given al, a2 E R \ {0} write al = a2gl + a3
with
a3 = 0 or v(a3) < v(a2)
a2 = a3q2 + a4
with
a4 = 0 or v(a4) < v(a3)
a3 = a4q3 + a5
with
a5 = 0 or v(a5) < v(a4)
Since v(a2) > v(a3) > v(a4) > , this process must terminate after a finite number of steps, i.e., an+1 = 0 for some n. For this n we have
an1 = angn1 + 0. Claim. an = gcd{al,a2}. Proof. If a, b E R then denote the gcd of {a, b} by the symbol (a, b). Theorem 5.5 shows that the gcd of {a, b} is a generator of the ideal generated by {a, b}. Now we claim that (a;, a;+1) = (at+1, ai+2) for 1 < i < n  1. Since a; = a;+1qi + ai+2i it follows that xa, + ya;+l = x(ai+lq, + ai+2) + yai+1 = (xqi + y)a;+1 + xa;+2
Thus, (a;, ai+1) S (at+1, ai+2), and similarly
ra;+1 + sat+2 = rat+i + s(a;  ai+lqi) = sat + (r  gi)at+1 so that (a;+1, at+2) C (at, ai+1) Hence, (a;, a;+1) = (at+1, ai+2), and we conclude (al, a2) = (a2, a3) = ... = (an1, an)
But (an,, an) = an since an (an1, and the claim is proved. This result gives an algorithmic procedure for computing the gcd of two elements in a Euclidean domain. By reversing the sequence of steps used to compute d = (a, b) one can arrive at an explicit expression d = ra + sb for the gcd of a and b. We illustrate with some examples. (5.20) Examples.
(1) We use the Euclidean algorithm to compute (1254, 1110) and write it as r1254 + x1110. Using successive divisions we get
Chapter 2. Rings
88
1254=1110.1+144 1110=144.7+102 144=102.1+42
102=42.2+18
42=18.2+6 18=6.3+0. Thus, (1254, 1110) = 6. Working backward by substituting into successive remainders, we obtain
6=4218.2 =42(10242.2).2=5.422.102
=54.125461.1110.
(2) Let f (X) = X2 X+ l and g(X) = X3 + 2X2 + 2 E Z3[X]. Then g(X) = Xf (X) + 2X + 2 and
f (X) = (2X + 2)2.
(3)
Thus, (f(X), g(X)) = (2X + 2) = (X + 1) and X + 1 = 2g(X) 2X f (X). We now give an example of how to solve a system of congruences in a Euclidean domain by using the Euclidean algorithm. The reader should
refer back to the proof of the Chinese remainder theorem (Theorem 2.24) for the logic of our argument. Consider the following system of congruences (in Z): x
1 (mod 15)
x=3 x
6
(mod 11) (mod 8).
We apply the Euclidean algorithm to the pair (88, 15):
88=15.5+13 15=13.1+2 13=2.6+1
2=1.2. Now we substitute backward to obtain
2.5 Principal Ideal Domains and Euclidean Domains
89
1=132.6 =15.6+(8815.5).7=88.715.41 =61615.41, so 616  1 (mod 15) and 616  0 (mod 88). Similarly, by applying the
Euclidean algorithm to the pair (120, 11), we obtain that 120  1 (mod 11) and 120  0 (mod 120), and by applying it to the pair (165, 8), we obtain 495  1 (mod 8) and 495 = 0 (mod 165). Then our solution is
x=I.(616)+3(120)+6(495)=3946 (mod 1320) or, more simply,
x  14 (mod 1320). We will now give some examples of Euclidean domains other than Z and FIX). If d 36 0,1 E Z is squarefree, we let Z[f = {a+b/ : a, b E Z}. Then Z[f is a subring of the field of complex numbers C and the quotient field of Z[f is the quadratic field
Q[f] ={a+bfd:a,bEQ}. (5.21) Proposition. If d E {2, 1, 2, 3 ) then the ring domain with v(a + bf) = Ia2  db21.
is a Euclidean
Proof. Note that v(a + brd) = Ia2  621 > 1 unless a = b = 0. It is a straightforward calculation to check that v(a/3) = v(a)v(Q) for every a, E Z[./al. Then
v(a/3) = v(a)v(0) > v(a), so part (1) of Definition 5.16 is satisfied. with /3 # 0. Then in the quotient field Now suppose that a, 0 E
of Z[f we may write a//3 = x + yf where x, y E Q. Since any rational number is between two consecutive integers and within 1/2 of the nearest integer, it follows that there are integers r, a E Z such that Ix  rl < 1/2
and ly  sI < 1/2. Let y = r + sv and b = /3((x  r) + (y  s)f). Then
a=/3(x+yv)=f3y+6. Since r, s E Z, it follows that y E Z[fd) and b = a  Qy E Z[ / also. Then
v(b) = v(p)v((x  r) + (y  s),Ad) = v(Q)I (x  r)2  d(y 
But
8)21.
Chapter 2. Rings
90
I(xr)2d(ys)2I < Ixrl2+Idlly812 < (1/2)2 + 3(1/2)2 = 1.
The only possibility for equality is when Ix  rl = ly  sl = 1/2 and d = 3. But in this case I(x  r)2  d(y  8)2I = 11/4  3/41 = 1/2 < 1.
Therefore, we always have I(x  r)2  d(y  8)21 < 1 and we conclude that v(6) < v(/3). Hence Z[vQ] is a Euclidean domain. Complementary to Proposition 5.21, we have the following result:
(5.22) Proposition. If d < 0 then Zlvfdl is a PID if and only if d = 1 or
d= 2. Proof. If d = 1 or d = 2 then by Proposition 5.21, Z1.1 rd] is a Euclidean domain and hence a PID. For the converse we need the following lemma.
(5.23) Lemma. 2 is not a prime in Z[fd).
Proof. Either d or d  1 is even so that 2 I d(d  1). But
d(d1)=d2d=(d+f)(dvd), so 21 (d+vd)(d v'd) but neither (d+ f)/2 nor (d v/ d)/2 are in Z[fJ. Thus 2 divides the product of two numbers, but it divides neither of the numbers individually, so 2 is not a prime element in the ring Z[f . We now return to the proof of Proposition 5.22. We will show that if d < 3 then 2 is an irreducible element of Z[f1. Since in a PID, irreducible and prime are the same concept (Corollary 5.7), it will follow that Z/ cannot be a PID because Lemma 5.23 shows that 2 is not a prime in Z[v' .
Suppose 2 = a/3 for a, /3 E Z['/d] with a and 3 not units. Then we must have v(a) > 1 and v(0) > 1. Therefore, 4 = v(2) = v(a)v($),
and since v(a), v(/3) E N it follows that v(a) = vL/3) = 2. Thus if 2 is not there is a number a = a + Wd E Z['//J such that irreducible in
v(a) = a2  db2 = ±2.
But ifd0+3.1>±2, while if b = 0 then a2  db2 = a2 96 ±2
2.5 Principal Ideal Domains and Euclidean Domains
91
when a E Z. Hence, if d < 3 there is no number a E Z[V' with v(c) = 2, Therefore, Proposition 5.22 and we conclude that 2 is irreducible in is proved.
(5.24) Remarks. (1)
A complex number is algebraic if it is a root of a polynomial with integer coefficients, and a subfield F C C is said to be algebraic if every element of F is algebraic. If F is algebraic, the integers of F are those elements of F that are roots of a monic polynomial with integer coefficients. In the quadratic field F = Q(v/31, every element of the ring Z[v/3) is an integer of F, but it is not true, as one might expect, that Z(v/3) is all of the integers of F. In fact, the following can be shown: Let d # 0, 1 be a squarefree integer. Then the ring of integers is of the field
Z[v' ] {Zr
if if
d
d
2, 3 (mod 4), 1 (mod 4).
l
In particular, the ring of integers of the field Q[vf3]l is the ring
R={a+bl
1
2:a,bEZ}.
We leave it as an exercise for the reader to prove that R is in fact a
Euclidean domain. (Compare with Proposition 5.22.) (2) So far all the examples we have seen of principal ideal domains have been Euclidean domains. Let R={a+b(1+219)
:a,bEZ} 11119
be the ring of integers of the quadratic field Q[ . Then it can be shown that R is a principal ideal domain but R is not a Euclidean domain. The details of the verification are tedious but not particularly
difficult. The interested reader is referred to the article A principal ideal ring that is not a Euclidean ring by J.C. Wilson, Math. Magazine (1973), pp. 3438. For more on factorization in the rings of integers of quadratic number fields, see chapter XV of Theory of Numbers by G. H. Hardy and E. M. Wright (Oxford University Press, 1960).
Chapter 2. Rings
92
2.6 Unique Factorization Domains We have seen in Section 2.5 that the fundamental theorem of arithmetic holds for any PID. There are, however, rings that are not PIDs for which the fundamental theorem of arithmetic holds. In this section we will give a result that allows for the construction of such rings.
(6.1) Definition. An integral domain R is a unique factorization domain (UFD) if every nonzero element a of R can be written essentially uniquely as a = upl pr where u is a unit and each pi is an irreducible element of R. Essentially uniquely means that if a = vql . q, where v is a unit and each q3 is irreducible, then r = 8 and, after reordering (if necessary), qi is an associate of pi. By collecting associate primes together, we may write (essentially uniquely) a=uipi'...pi+4
where pi is not an associate of p_, if i A j. This is called the prime factorisation of a and the primes pi are called the prime factors or prime divisors of a.
(6.2) Lemma. Let R be a UFD.
(1) An element a E R is irreducible if and only if it is prime. (2) Any nonempty set of nonzero elments of R has a greatest common divisor.
Proof. (1) Suppose a E R is irreducible and a I be. Thus ad = be for some d E R. Writing b, c, and d as a product of units and irreducible elements gives
auldl ... dr = u2b1 ... b,u3c1 ... ct where each bi, c,, and dk is irreducible and each ui is a unit. By uniqueness of factorization of be, it follows that the irreducible element a is an associate of some b, or c2 and, hence, a I b or a I c. (2) We prove this in the case that the set in question is {a, b} consisting
of two elements and leave the general case for the reader. Let pl, .... pr denote all the primes that are a prime factor of either a or b. Then we may write a=Pi'...Pr
and b=Pi'...pr
where 0 < mi and 0 < n, for each i. Let ki = min{mi,ni} for l < i < r and let
d=pii ...pr'
We claim that d is a gcd of a and b. It is clear that d I a and d I b, so let e be any other common divisor of a and b. Since e I a, we may write a = ec
2.6 Unique Factorization Domains
93
for some c E R. Taking prime factorizations of e, a, and c and applying the unique factorization assumption, we conclude that any pri.ae factor of e must also be a prime factor of a and the power of the prime that divides e can be no more than the power that divides a. Thus, since e also divides b, we may write
e = pi' ...
p
where P; < min{mi, ni} = ki. Therefore, every prime factor of e is also a prime factor of d and the power of the prime factor dividing e is at most 0 that which divides d. Thus e I d and d is a gcd of a and b. Our goal is to prove that if R is a UFD then the polynomial ring R(X] is also a UFD. This will require some preliminaries.
(6.3) Definition. Let R be a UFD and let f (X) 0 0 E R(X]. A gcd of the coefficients of f (X) is called the content of f (X) and is denoted cont(f (X)). The polynomial f (X) is said to be primitive if cont(f (X)) = 1.
Note that the content of f (X) is only uniquely defined up to multiplication by a unit of R. If f (X) is a nonzero polynomial then we can write f (X) = c f1(X) where f, (X) is primitive and c is the content of f (X).
(6.4) Lemma. (Gauss's lemma) Let R be a UFD and let f (X), g(X) be nonzero polynomials in R(X]. Then cont(f (X)g(X )) = cont(f (X)) cont(g(X)).
In particular, if f (X) and g(X) are primitive, then the product f (X)g(X ) is primitive.
Proof. Write f(X) = cont(f(X))fi(X) and g(X) = cont(g(X))91(X) where f1(X) and gi (X) are primitive polynomials. Then
f (X)g(X) = cont(f (X)) cont(g(X)) fl(X)gj(X), so it is sufficient to check that f1(X)g1(X) is primitive. Now let
fi(X) =ao+a1X +...+anX'n and
gl(X) and suppose that the coefficients of fl (X )gi (X) have a common divisor d other than a unit. Let p be a prime divisor of d. Then p must divide all of the coefficients of fi (X)g1(X ), but since f1(X) and gi (X) are primitive, p does not divide all the coefficients of f, (X) nor all of the coefficients of gi (X). Let a, be the first coefficient of fi (X) not divisible by p and let b, be the first coefficient of g1(X) not divisible by p. Consider the coefficient of X'+° in f i (X )gi (X ). This coefficient is of the form
Chapter 2. Rings
94
arbs + (ar+lba1 + ar+2bs2 +
) + (arlbs+1 + ar2ba+2 + ' )
By hypothesis p divides this sum and all the terms in the first parenthesis are divisible by p (because p I bi for j < a) and all the terms in the second parenthesis are divisible by p (because p I aj for i < r). Hence p I arb, and since p is prime we must have p I ar or p I b contrary to our choice of ar and b,. This contradiction shows that no prime divides all the coefficients of f, (X)gl (X), and hence, f, (X)gl (X) is primitive. 0 (6.5) Lemma. Let R be a UFD with quotient field F. If f (X) 34 0 E F[X], then f (X) = of, (X) where a E F and f, (X) is a primitive polynomial in R[X]. This factorization is unique up to multiplication by a unit of R. Proof. By extracting a common denominator d from the nonzero coefficients
of f (X), we may write f (X) = (1 /d) f (X) where f (X) E R(X]. Then let a = cont(f(X))/d = c/d E F. It follows that f(X) = of, (X) where f1(X) is a primitive polynomial. Now consider uniqueness. Suppose that we can also write f (X) =,3f2(X) where f2 (X) is a primitive polynomial in R(X ] and Q = a/b E F. Then we conclude that (6.1)
adf2(X) =cbf1(X)
The content of the left side is ad and the content of the right side is cb, so ad = ucb where u E R is a unit. Substituting this in Equation (6.1) and dividing by cb gives
uf2(X) = f1(X) Thus the two polynomials differ by multiplication by the unit u E R and 0 the coefficients satisfy the same relationship 3 = a/b = u(c/d) = ua. (6.6) Lemma. Suppose R is a UFD with quotient field F. If f(X) E R[X] has positive degree and is irreducible in R[X], then f(X) is irreducible in FIX]. Proof, If f (X) E R[X] has positive degree and is irreducible in R(X ] then J (X) is primitive since cont(f (X)) I f (X) in R(X]. Suppose that f (X) is reducible in FIX]. Thus f (X) = g1(X)g2(X) where g;(X) E F[X] and
degg;(X) > 0 for i = 1 and 2. Then g;(X) = a;f;(X) where a, E F and f1(X) E R[X] is primitive. Thus,
P X) = a1a2f1(X)f2(X ),
and the product f1(X)f2(X) is primitive by Gauss's lemma. Thus, by Lemma 6.5, f(X) and f1(X)f2(X) differ by multiplication by a unit of R, which contradicts the irreducibility of f (X) in R[X). Thus, we conclude 0 that f (X) is irreducible in F(X].
2.6 Unique Factorization Domains
95
(6.7) Corollary. Let R be a UFD with quotient field F. Then the irreducible elements of R[X] are the irreducible elements of R and the primitive polynomials f(X) E R[X] which are irreducible in FIX]. Proof.
(6.8) Theorem. If R is a UFD, then R[X] is also a UFD.
Proof. Let F be the quotient field of R and let f(X) 96 0 E R[X]. Since F[X] is a UFD, we can write
f(X) =P1(X) ... pr(X) where pi(X) E F[X] is an irreducible polynomial for 1 < i < r. By Lemma
6.5, pi(X) = aigi(X) where a; E F and qi(X) E R[X] is a primitive polynomial. Thus, we have
f(X)
=cg1(X)...gr(X)
where c = al . ar E F. Write c = a/b where a, b E R. Then taking contents we get cont(bf (X)) = cont(agi (X) . . . qr(X)) = a
by Gauss's lemma. Thus, b cont (f (X)) = a, so b ( a and cont (f (X)) = c E R. Each qi(X) is irreducible in F[X], and hence, it is irreducible in R[X].
Since R is a UFD, write c = udi ... d, where each di is prime in R and u E R is a unit. Thus we have a factorization
f(X) = udi...d9g1(X)...gr(X) of f(X) into a product of irreducible elements of R[X]. It remains to check uniqueness. Thus suppose we also have a factorization
f(X) = vbi ...btq'(X)...gk(X) where each q;(X) is a primitive polynomial in R[X] and each bi is an irreducible element of R. By Corollary 6.7, this is what any factorization into irreducible elements of R[X] must look like. Since this also gives a factorization in FIX] and factorization there is unique, it follows that r = k and qi(X) is an associate of q;(X) in FIX] (after reordering if necessary). But if primitive polynomials are associates in FIX], then they are associates in R[X] (Lemma 6.5). Furthermore,
cont(f(X)) =vbi ...bt =ud1 ... d so s = t and bi is an associate of d, (after reordering if necessary) since R is a UFD. This completes the proof.
96
Chapter 2. Rings
(6.9) Corollary. Let R be a UFD. Then R[X,, ... , X,,J is also a UFD. In particular, FIX,, ... , XnJ is a UFD for any field F.
0
Proof. Exercise.
(6.10) Example. We have seen some examples in Section 2.5 of rings that
are not UFDs, namely, F[X2, X3J and some of the quadratic rings (see Proposition 5.22 and Lemma 5.23). We wish to present one more example of a Noetherian function ring that is not a UFD. Let
S1={(x,y)ER2:z2+y2=1} be the unit circle in R2 and let I C R[X, YJ be the set of all polynomials such that f (x, y) = 0 for all (x, y) E S'. Then I is a prime ideal of R[X, Y) and R[X, YJ/I can be viewed as a ring of functions on S' by means of
f (X, Y) +I i f where 7(x, y) = f (x, y). We leave it for the reader to check that this is well defined. Let T be the set of all f (X, Y) + I E R[X, YJ/I such that 7(x, y) j4 0 for all (x, y) E S'. Then let the ring R be defined by localizing at the multiplicatively closed set T, i.e., R = (R[X, YJ/I )T .
Thus, R is a ring of functions on the unit circle, and a function in R is a unit if and only if the function never vanishes on S1. Claim. R is not a UFD. Proof. Let g(x, y) = x2 + (y  1)2 and h(x, y) = x2 + (y + 1)2 E R. Then
gh(x,y) = (x2 + (y  1)2)(x2 + (y + 1)2) x4 + x2((y  1)2 + (y + 1)2) + (y2 = = x4 + x2((y  1)2 + (y + 1)2) + x4
 1)2
so that x divides gh in R, but clearly x does not divide either g or h (since neither g nor h is zero at both (0, 1) and (0, 1) E S', but x is). Therefore the ideal (x) C R is not prime. The proof is completed by showing
that x is an irreducible element of the ring R. To see this suppose that x = f 1 f 2 where neither is a unit. Then we must have V(fl) = 1(0, 1)) and V(f2) = { (0, 1) } (or vice versa), where V (g) means the set of zeros of the function g. Since fl and f2 are continuous functions, it follows that fl does not change sign on S1 \ {(O, 1) } and f2 does not change sign on S' \ ((0, 1)).
Therefore, x = fl f2 will not change sign on the set S1 \ 1(0, 1), (0, 1)}, and this contradiction shows that x must be an irreducible element of R. 0
2.6 Unique Factorization Domains
97
In general, the problem of explicitly determining if a polynomial in R[X] is irreducible is difficult. For polynomials in Z[X] (and hence in Q[X])
there is a procedure due to Kronecker that in principle can determine the factors of an integral polynomial in a finite number of steps. For this method see, for example, Modern Algebra by B.L. van der Waerden, Vol. I, p. 77. We shall content ourselves with the following simple criterion for irreducibility, which is sometimes of use. An example of its use will be in Corollary 6.12.
(6.11) Theorem. (Eisenstein's criterion) Let R be a UFD with quotient field
F. Let f (X) = ao + a1X +
+ anX'i (an 0 0) be in R[X] and suppose
that p E R is a prime such that p does not divide an,
pIai for 0 Si 1 in FIX] then by Lemma 6.6 there is also a factorization in R[X]. Thus, suppose that we can
write f (X) = g(X)h(X) where g(X), h(X) E R[X]. From Gauss's lemma we must have that g(X) and h(X) are primitive polynomials. Suppose
with f, m > 1, btc,,, 0, and e + m = n. Since p I ao = b0co but p2 does not divide a0, it follows that p I b0 or p I co but not both. To be specific, suppose that p I b0 but that p does not divide co. Not all the coefficients of g(X) are divisible by p since g(X) is primitive. Let bi be the first coefficient of g(X) that is not divisible by p so that 0 < i < t < n. Then we have a; = bico + bi_ic1 + ... + b0ci.
But p I a, and p I bj for j < i, so p I bic0. But p does not divide bi and p does not divide co, so we have a contradiction. Therefore, we may not
write f (X) = g(X)h(X) with deg g(X) > 1, deg h(X) > 1, i.e., f (X) is irreducible in FIX]. 0 The following is a useful consequence of Eisenstein's criterion:
(6.12) Corollary. Let p be a prime number and let f9(X) E Q[X] be the polynomial
98
Chapter 2. Rings
XP1
fp(X) = X P1 + XP2 + ... + X + l = X1 Then fp(X) is irreducible.
Proof. Since the map g(X) ' g(X + 1) is a ring isomorphism of Q[X], it is sufficient to verify that fp(X + 1) is irreducible. But p (by Exercise 1) divides every binomial coefficient (k) for 1 < k < p, and hence, P 
fP(X+1)= (X+l1)1
XP1+pXP2+...+p.
Thus, fp(X + 1) satisfies Eisenstein's criterion, and the proof is complete. 0
2.7 Exercises 1.
2.
Prove the binomial theorem (Proposition 1.12). Give a counterexample if a and b do not commute. If p is a prime number, prove that the binomial coefficient (k) is divisible by p for 0 < k < p. Give an example to show that this result need not be true if p is not prime. Let R be a ring with identity and let a E R. The element a is said to be nilpotent if an = 0 for some n E N. It is said to be idempotent if a2 = a. Prove the following assertions.
3.
4.
5. 6. 7.
8. 9.
(a) If R has no zero divisors, then the only nilpotent element of R is 0 and the only idempotent elements of R are 0 and 1. b) No unit of R is nilpotent. The only idempotent unit of R is I. (c) If a is nilpotent, then 1  a is a unit. (Hint: Geometric series.) If a is idempotent, then 1  a is idempotent. (d) If R is commutative and N = {a E R : a is nilpotent}, show that N is an ideal of R. (e) Provide a counterexample to part (d) if R is not commutative. (Note that in parts (b)(e), the ring R is allowed to have zero divisors.) For the ring P(X) of Example 1.10 (7), show that every A E P(X) satisfies the equation A2 = A. If P(X) is an integral domain, show that IXI = 1.
Continuing with the ring P(X), let a E X and define I. = A E P(X) a Al. Prove that la is a maximal ideal of P(X). What is P X)/I,? For a finite set X determine all of the ideals of P(X) and show that every maximal ideal is an ideal I. for some a E X. Prove Lemma 2.12 (1) and (3). Let R be a ring with identity. Show that R is a division ring if and only if it has no left or right ideals other than {0} and R. a Solve the equation 6x = 7 in the ring Z19, if possible. (b) Solve the equation 6x = 7 in the ring Z2o, if possible.
If R and S are rings with identities, prove that (R x S)' = R' x S. (Recall that R' denotes the group of units of the ring R.) Compute all the ideals, prime ideals, and maximal ideals of the ring Z60. What are the nilpotent elements of Z6e?
2.7 Exercises 10.
99
Let R be a ring and let R°P ("op" for opposite) be the abelian group R, together with a new multiplication a b defined b a b = ba, where ba
denotes the given multiplication on R. Verify that R°" is a ring and that the identity function 1R : R R°P is a ring homomorphism if and only if R is a commutative ring. 11.
(a) Let A be an abelian group. Show that End(A) is a ring. (End(A) is defined in Example 1.10 (11).) (b) Let F be a field and V an Fvector space. Show that EndF(V) is a ring. Here, Endp(V) denotes the set of all Flinear endomorphisms of V, i.e.,
EndF(V)={hEEnd(V):h(av)=ah(v)forallvEV,aEF}. In this definition, End(V) means the abelian group endomorphisms of V,
12.
and the ring structure is the same as that of End(V) in part (a). (a) Let R be a ring without zero divisors. Show that if ab = 1 then ba = 1 as well. Thus, a and b are units of R. (b) Show that if a, b, c E R with ab = 1 and ca = 1, then b = c, and thus a (and b) are units. Conclude that if ab = 1 but ba $ 1, then neither a nor b are units. (c) Let F be a field and FIX] the polynomial ring with coefficients from F. F[X] is an Fvector space in a natural way, so by Exercise 11, R = EndF(FF[X]) is a ring. Give an example of a, b E R with ab = 1 but ba54 1.
13.
(a) Let x and y be arbitrary positive real numbers. Show that the quaternion
algebras Q(x, y; R) and Q(1, 1; R) are isomorphic. (b) Show that the quaternion algebras Q(1, 3;Q), Q(1, 7;Q), and Q(1, 11; Q) are all distinct.
(c) Analogously to Example 1.10 (10), we may define indefinite quaternion algebras by allowing x or y to be negative. Show that, for any nonzero
real number x and any subfield F of R, Q(1, x; F) and Q(l, x; F) are isomorphic.
(d) Show that for any nonzero real number x, Q(1, x; R) and Q(1, 1; R) are isomorphic.
14. 15.
(e) Show that for any subfield F of R, Q(1, 1; F) is isomorphic to M3(F), the ring of 2 x 2 matrices with coefficients in F. (Thus, Q(1, 1; F) is not a division ring.) Verify that Z(i)/(3 + i) Zio. Let F be a field and let R C F(XJ x F[Y] be the subring consisting of all pairs (f (X), g(Y)) such that f (0) = g(0 . Verify that FIX, YJ/(XY) 5 R.
16.
Let R be a ring with identity and let I be an ideal of R. Prove that
17.
Let F be a field and let R = { [ o 0] E M2 (F) 1. Verify that R is a ring. Does
5_
R have an identity? Prove that the set I = { o o] E R} is a maximal ideal of R. 18.
19.
(a) Given the complex number z = I + i, let o : R(X) . C be the substitution homomorphism determined by z. Compute er(m). (b) Give an explicit isomorphism between the complex numbers C and the quotient ring R[XJ/(X'  2X + 2). Let R = C11[0, 1]) be the ring of continuous realvalued functions on the interval (0,. Let T C [0, 11, and let
I(T)={f E R: f(x)=0forallxET}. a Prove that I(T) is an ideal of R. b If x E 10, 1] and M. = I({x}), prove that M. is a maximal ideal of R and R`M= ?f R.
100
Chapter 2. Rings (c) If S C R let
V(S)=(xEI: f(s)=0forall f ES). Prove that S is a closed subset of 10,1]. (You may quote appropriate
theorems on continuous functions.)
(d) If I R is any ideal, then prove that V(1) j4 0. (Hint: Suppose that V(I) = 0 and construct a function f E I such that f (x) 96 0 for all x E [0,1 . Conclude that I = R. At some point in your argument you
will need to use the compactness of 0,11.) (e) Prove that any maximal ideal M of is M. for some x E 10, 1]. 20. Let R C S be commutative rings with identity and let d E R be an element
such that the equation a2 = d is solvable in S but not in R, and let f denote a solution to the equation in S. Define a set R[ fd] by
R[vd]=(a+bf:a,bER) CS. (a) Prove that R[fd] is a commutative ring with identity. (b) Prove that Z[d] is an integral domain. (c) If F is a field, prove that F[fd] is a field. 21.
(a) If R = Z or R = Q and d is not a square in R, show that R[ fd] °_° R[X ]/ (X 2  d) where (X2  d) is the principal ideal of R(X] generated by X2  d. (b) If R = Z or R = Q and di, d2, and di/d2 are not squares in R \ {0}, show that R[ f j ] and R[ v/ 2 are not isomorphic.
(The most desirable proof of these assertions is one which works for both R = Z and R = Q, but separate proofs for the two cases are acceptable.)
(c) Let R, = ZD[X]/(X2  2) and R2 = ZD[X]/(X2  3). Determine if
Rl°°R2in case p=2,p=5,or p= 11.
22.
Recall that R' denotes the group of units of the ring R. (a) Show that (Z(vr1])' = {±1,
(b) If d < 1 show that (Z[f])' _ {t1). c) Show that
Z((1+" ] =ltl }1+ 2
2
l+ 2
(d) Let d > 0 E Z not be a perfect square. Show that if Z(d) has one unit other than ±1, it has infinitely many. (e) It is known that the hypothesis in part (d) is always satisfied. Find a unit in Z[%/dl other than ±1 fcr 2 < d < 15, d A 4, 9. 23. Let F Q be a field. An element a E F is said to be an algebraic integer if for some monic polynomial p(X) E Z[X], we have p(a) = 0. Let d E Z be a nonsquare.
(a) Show that if a E F is an algebraic integer, then a is a root of an irreducible monic polynomial p(X) E Z[X]. (Hint: Gauss's lemma.)
(b) Verify that a + bf E Q[fd] is a root of the quadratic polynomial p(X) = X2  2aX + (a2  bed) E Q[X]. (c) Determine the set of algebraic integers in the fields Q[f] and Q(f]. 24.
(See Remark 5.24 (1).) If R is a ring with identity, then Aut(R), called the automorphism group of R, denotes the set of all ring isomorphisms m : R R.
2.7 Exercises
101
(a) Compute Aut(Z) and Aut(Q). b Compute Aut(Z[v' J) and Aut Q 'Q[X11) if d is not a square in Z or Q. ' Q[X] be the substitution homo(c) If a j4 0 E Q,b E Q let 0. s : morphism determined by V '# aX + b. Prove that 4,,6 E Aut(Q(X]).
What is 0?
(d) If 0 E Aut(Q[XI) prove that there are a # 0, b E Q such that 0 = O,,s. 25. Let m = exp(2ai/n) E C and let R = Q[w). Show that Aut(R) °_e Z. 26.
Let R be a commutative ring with identity. Prove that Mn(R[X]) ?°
27.
Mn(R)[X] Let R be a commutative ring with identity and let f = ao + a1X +
+ a,X" E R[X] where an 96 0. We know that if R is an integral domain then the units of R[X] are the units of R (Corollary 4.3). This exercise will
investigate when f is a unit, a zero divisor, or is nilpotent when R is not assumed to be an integral domain.
(a) Prove that f is a unit in R[X) if and only if ao is a unit in R and
+ b,nX'n is the inverse a1, ... , an are nilpotent. (Hint: If bo + biX + of f, prove by induction on r that a'n+lb,_,. = 0. Conclude that an is nilpotent and apply Exercise 2 (c).) (b) Prove that f is nilpotent if and o if ao, al , ... , an are nilpotent. c Prove that f is a zero divisor in R[X] if and only if there is a nonzero
a E R such that of = 0. (Hint: Choose a nonzero polynomial g =
+ b,. X"' of least degree m such that f = 0. If m = 0 we are done. Otherwise, anb,n = 0 and hence ang = 0 (because (an9)f = an (g f) = 0 and deg(ang) < degg). Show by induction that an,.g = 0 for (0 < r < n). Conclude that 6,,, f = 0.) 28. Factor the polynomial X2 + 3 into irreducible polynomials in each of the bo + bl X +
following rings.
a Z5 X.
(b) 29.
1X I.
Let F = Z5 and consider the following factorization in FIX]:
3X3+4X2+3=(X+2)2 (3X+2) _ (X + 2)(X + 4)(3X + 1).
Explain why (*) does not contradict the fact that F[X] is a UFD. 30. For what fields ZD is X 3 + 2X2 + 2X + 4 divisible by X 2 + X + I? 31. In what fields Zy is X2 + 1 a factor of X 3 + X 2 + 22X + 15? 32. Find the gcd of each pair of elements in the given Euclidean domain and mess the gcd in the form ra + aA .
a 189 and 301 in Z.
(b 1261 and 1649 in Z.
(c) X4X3+4X2X+3 and X32X2 +X2 inQ[X]. d) X4 + 4 and 2X3 + X2  2X  6 in Z3 [X ].
e 2 + IN and 1 + 3i in Z[i].
f) 4 + 7i and 1 + 7i in Z(i]. 33. Express X4  X3  2 as a product of irreducible polynomials in each of the fields Q, 11, C, and Z5. 34.
Let F be a field and suppose that the polynomial X'n  1 has m distinct roots in F. If k [ m, prove that X"  1 has k distinct roots in F.
35.
If R is a commutative ring, let F(R, R) be the set of all functions f : R + R and make F(R, R) into a ring by means of addition and multi lication of functions, i.e.,
function 4ti : R[
g)(r) = f(r) g(r) and (f + 9)(r) = f (r) + g(r). Define a ]  (R, ) by
102
Chapter 2. Rings
4,(f (X)) (r) = Or(f (X)) = f (r) for all r E R. a Show that 40 is injective if R is an infinite integral domain. b Show that 0 is not injective if R is a finite field. 36. 37
Let f (X) E Z [X]. Show that f (X') = (f (X ))P and that the map O(f M) = (f OF W is a ring homomorphism. Let F be a field and consider the substitution homomorphism
m: FIX,YI .FIT] such that m(X) = T2 and O(Y) = T3. Show that Ker(O) is the principal ideal generated by y2  X3. What is Im(m)? 38.
Prove Proposition 4.10 (Lagrange interpolation) as a corollary of the Chinese remainder theorem.
39.
Let F be a field and let ao, ai, ... , an be n + 1 distinct elements of F. If f : F  F is a function, define the successive divided differences of f, denoted f [ao, ... , a,], by means of the inductive formula: flao[ = f(ao)
flao, all =
f[ao,...,an]=
f[ai]  flan] al  ao
f[ao, ... ,an2, an]  f[ao, ... ,an2, ani] an  ani
Prove that the coefficient of Pi (X) in Equation (4.3) (Newton's interpolation formula) is the divided difference f [ao, ... , a,]. 40. (a) If f (X) E RIX] and z E C satisfies f (z) = 0, show that f ('z) = 0 (where a is the complex conjugate of z). (b) Let ro, ri, ..., rk be distinct real numbers, let zi, ... , z,,, be distinct complex numbers with z., All f o r 1 < j, f < m, and let 80, ... , sk E R,
Wi, ... , wn, E C. Prove that there is a polynomial f (X) E R[X] such
that f(r,) = s; for 0 < i < k, while f(zj) = ws for 1 < j < m. What degree can we require for 1(X) in order to have a uniqueness statement as in Proposition 4.10? 41.
Let F be a field and let f,g E FIX] with degg > 1. Show that there are unique fo, f i ,
, fd E FIX I such that deg f, < deg g and
f
=fo+fig+...+fd9d.
Let K and L be fields with K C L. Suppose that f (X), g(X) E K[X). (a) If f (X) divides g(X) in L[X], prove that f (X) divides g(X) in XJ. b Prove that the greatest common divisor of f (X) and g(X) in KJX is the same as the greatest common divisor of f (X) and g(% in L X . (We will always choose the monk generator of the ideal (f (X), g(X) as the greatest common divisor in a polynomial ring over a field.) 43. (a) Suppose that R is a Noetherian ring and I C R is an ideal. Show that Rh is Noetherian. (b) If R is Noetherian and S is a subring of R, is S Noetherian? c) Suppose that R is a commutative Noetherian ring and S is a nonempty multiplicatively closed subset of R containing no zero divisors. Prove that Rs (the localization of R away from S) is also Noetherian. 42.
2.7 Exercises 44.
103
If R is a ring (not necessarily commutative) and f (X) = ao + aiX + + a,, X" E R[X], then we say that f (X) is regular o degree n if an is a unit of R. Note, in particular, that monic polynomials are regular and if R is a field then all nonzero polynomials in R[X) are regular. Prove the following version of the division algorithm:
Let f(X) E R(XJ and let g(X) E R[X] be a regular polynomial of degree n. Then there are unique polynomials ql (X ), ri (X), q2 (X ), and r2 (X) E R[X] such that deg rl (X) < n, deg r2 (X) < n,
f(X) = gi(X)g(X) +rl(X) and
f(X) =g(X)g2(X)+r2(X) The two equations represent the left and right divisions of f (X) by 9(X). In the special case that g(X) = X  a for a E R, prove the following version of these equations (noncommutative remainder theorem):
+ anX" E R[X] and let a E R. Then there are
Let f (X) = a1 + a1X +
unique qc(X) and qR (X) E R(X ] such that
f(X) = gR(X)(X  a) + fx(a) and
f(X) = (X a)gc(X)+ft(a) where
n
fn (a) = 1:
n
akak
akak
and f c (a) _
k=0
k=0
are, respectively, the right and left evaluations of f (X) at a E R. (Hint: Use the formula Xk
 ak = (X k1 + X k2a + ... + Xak2 + ak1) (X  a) = (X  a)(Xk1 + aXk2 + ... + ak2X + ak1).
Then multiply on either the left or the right by at, and sum over k to get the division formulas and the remainders.) 45. 46.
47.
Let R be a UFD and let a and b be nonzero elements of R. Show that ab = (a, bJ(a, b) where [a, b) = lcm{a, b} and (a, b) = gcd{a, b}. Let R be a UFD. Show that d is a gcd of a and b (a, b E R \ {0}) if and only if d divides both a and b and there is no prime p dividing both a/d and b/d. In particular, a and b are relatively prime if and only if there is no prime p dividing both a and b.)
Let R be a UFD and let {r1}1 be a finite set of pairwise relatively prime nonzero elements of R (i.e., ri and r3 are relatively prime whenever i 36 j).
Let a = f 1 ri and let a, = a/ri. Show that the set {ai} 1 is relatively prime.
Let R be a UFD and let F be the quotient field of R. Show that d E R is a square in R if and only if it is a square in F (i.e., if the equation a2 = d has a solution with a E F then, in fact, a E R). Give a counterexample if R is not a UFD. 49. Let x, y, z be integers with gcd x, y, z = 1. Show that there is an integer a such that cd{x + ay, z} = 1. (Hint: The Chinese remainder theorem may be helpful.) 48.
104
50.
51.
Chapter 2. Rings
According to the Chinese remainder theorem there is an isomorphism of rings ¢ : Z60  Z3 X Z4 X Z5. Compute 0(26), 4)(35), 41(2, 3, 4), and m1(1, 2, 2). Solve the system of simultaneous congruences:
3 (mod 13) x  16 (mod 18) x 2 (mod 25) x
x=0 52.
(mod 29).
Solve the system of simultaneous congruences:
x6 (mod21)
x9 (mod33) x
2
(mod 37).
53. For what values of a (mod 77) does the following system of simultaneous
congruences have a solution?
x6 (mod21) x  9 (mod 33) x =a (mod 77). 54.
(a) Solve the following system of simultaneous congruences in Q(XI:
f(X) f (X) f (X)
3 (mod X + 1) 12X
4X
(mod X2  2) (mod X3).
b) Solve this system in Z5 X . c) Solve this system in Z3 X . d Solve this system in Z2 X . 55. Suppose that ml, m2 E Z are not relatively prime. Then prove that there are integers al, a2 for which there is no solution to the system of congruences:
x  a1 (mod ml) x  a2 (mod M2)56.
Let R be a UFD. Prove that
f(X,Y)=X4+2Y2X3+3Y3X2+4YX+5Y+6Y2 is irreducible in the polynomial ring R(X, Y].
Prove that if R is a UFD and if j(X) is a monic polynomial with a root in the quotient field of R, then that root is in R. (This result is usually called the rational root theorem.) 58. Suppose that R is a UFD and S C R\ {0} is a multiplicatively closed subset. Prove that Rs is a UFD. 59. Let F be a field and let F((X)j be the ring of formal power series with coefficients in F. If j = F,°° u a X" 0 0 E F, let o(f) = min{n # 0} and define o(0) = oo. o(f) is usually called the order of the power series j. 57.
Prove the following facts:
(a) o(f9) = o(f) + o(9).
2.7 Exercises
105
b) of +g) >_ min{o(f), o(g) }.
). cd ffis9ifandonlyifo(f)anX'
F,m E Z is arbitrary
:a
n=m
where the ring operations are defined as for F[[X]]. Prove that F((X)) is isomorphic to the quotient field of the integral domain F[IX]]. 61.
Let R be a commutative ring and let S = R[T1 , ... , Tn]. Define f (X) E S[X] by n
f(X) = ll(X Ti) i=1 n
= (1)ror(Ti, ... Tn)Xf
r.
r=0
(a) Show that
Tit ...Tir.
ar(T1, ... Tn) = Thus, a, = Ti +
+Tn and on = Ti .. Tn. or is called the r`h elemen
tary symmetric function. Therefore, the coefficients of a polynomial are obtained by evaluating the elementary symmetric functions on the roots of the polynomial.
(b) If g(X) = [I,=, (1 T;X), show that the coefficient of Xr is i.e., the same as the coefficient of X"r in f(X). (c) Define sr(T1, ... Tn) = T; + + Tn for r > 1, and let so = n. Verify the following identities (known as Newton's identities) relating the power sums sr and the elementary symmetric functions ok. r1
(1)kaksrk + (1)rror
=0
(1 < r < n)
k=0 n
J( I)kakSrk = 0
(r > n).
k=0
(Hint: Do the following calculation in S[[X]], where' means derivative with respect to X: g' (X)
g(X)
n n
i=s
1TX
00
i=1 i=o
00
ELT'+1Xj Esj+1Xj. i=o
Now multiply by g(X) and compare coefficients of X `1 . )
Chapter 2. Rings
106
62.
Let P E Z be a prime and define
R = {a = (al, as, a3,
.
.) :
ak E (Z/pkZ), ak+1 = ak
(mod pk )
for all
k>1).
(a) Show that R is a ring under the operations of componentwise addition and multiplication. (b) Show that R is an integral domain. (Note that R contains Z as a subring so that char(R) = 0.) (c) Let
P={aER:a1=0EZ/pZ}.
Show that every element of R \ P is invertible. Show that P is a proper ideal of R. (Thus, P is the unique maximal ideal in R and so R is a local ring.)
(d) For a E R \ {0} define v(a) to be p^' 1 if n is the smallest value of k such that ak j4 0 E Z/pkZ. Show that v : R \ {0} + Z+ makes R into a Euclidean domain.
Remark. The ring R plays an important role in mathematics; it is known as the ring of padic integers and its quotient field is known as the field of padic numbers.
Chapter 3
Modules and Vector Spaces
3.1 Definitions and Examples Modules are a generalization of the vector spaces of linear algebra in which the "scalars" are allowed to be from an arbitrary ring, rather than a field. This rather modest weakening of the axioms is quite far reaching, including, for example, the theory of rings and ideals and the theory of abelian groups as special cases.
(1.1) Definition. Let R be an arbitrary ring with identity (not necessarily commutative). (1)
A left Rmodule (or left module over R) is an abelian group M together with a scalar multiplication map
:RxMM that satisfy the following axioms (as is customary we will write am in place of (a, m) for the scalar multiplication of m E 1LI by a E R). In these axioms, a, b are arbitrary elements of R and m, n are arbitrary elements of M.
(ai)a(m + n) = am + an. (bi)(a + b)m = am + bm. (ci) (ab)m = a(bm).
(di)lm = m.
(2) A right Rmodule (or right module over R) is an abelian group M together with a scalar multiplication map
:MxR+M that satisfy the following axioms (again a, b are arbitrary elements of R and m, n are arbitrary elements of M).
(&)(m + n)a = ma + na. (b,)m(a + b) = ma + mb.
108
Chapter3. Modules and Vector Spaces
(cr)m(ab) = (ma)b. (dr)m1 = m.
(1.2) Remarks. (1) If R is a commutative ring then any left Rmodule also has the struc
ture of a right Rmodule by defining mr = rm. The only axiom that requires a check is axiom (cr). But m(ab) = (ab)m = (ba)m = b(am) = b(ma) = (ma)b.
(2) More generally, if the ring R has an antiautomorphism (that is, an additive homomorphism 0: R ' R such that 0(ab) = m(b)4(a)) then any left Rmodule has the structure of a right Rmodule by defining ma = ¢(a)m. Again, the only axiom that needs checking is axiom (cr):
(ma)b = .(b)(ma) = O(b)(O(a)m) = (m(b)O(a))m
= 0(ab)m
= m(ab).
An example of this situation occurs for the group ring R(G) where R is a ring with identity and G is a group (see Example 2.1.10 (15)). In this case the antiautomorphism is given by
O(Ea99) _ 9EG
a99' 9EG
We leave it as an exercise to check that : R(G)  R(G) is an antiautomorphism. Thus any left R(G)module M is automatically a right R(G)module.
(3) Let R be an arbitrary ring and let R°P ("op" for opposite) be the ring whose elements are the elements of R, whose addition agrees with
that of R, but whose multiplication  is given by a b = ba (where the multiplication on the righthand side of this equation is that of R). Then any left Rmodule is naturally a right R°Pmodule (and viceversa). In fact, if M is a left Rmodule, define a right multiplication
of elements of R°P (which are the same as elements of R) on M by am. As in Remark 1.2 (1), the only axiom that requires checking is axiom (Cr). But
The theories of left Rmodules and right Rmodules are entirely parallel, and so, to avoid doing everything twice, we must choose to work on
3.1 Definitions and Examples
109
one side or the other. Thus, we shall work primarily with left Rmodules unless explicitly indicated otherwise and we will define an Rmodule (or module over R) to be a left R module. (Of course, if R is commutative, Remark 1.2 (1) shows there is no difference between left and right Rmodules.) Applications of module theory to the theory of group representations will, however, necessitate the use of both left and right modules over noncommutative rings. Before presenting a collection of examples some more notation will be introduced.
(1.3) Definition. Let R be a ring and let M, N be Rmodule8. A function f : M + N is an Rmodule homomorpbism if
(1) A MI + m2) = A MO + P M2) for all ml, m2 E M, and (2) f (am) = a f (m) for all a E R and m E M.
The set of all Rmodule homomorphisms from M to N will be denoted HomR(M, N). In case M = N we will usually write EndR(M) rather than HomR(M, M); elements of EndR(M) are called endomorphisms. If f E EndR(M) is invertible, then it is called an automorphism of M. The group of all Rmodule automorphisms of M is denoted AutR(M) (Aut(M)
if R is implicit). If f E HomR(M, N) then we define Ker(f) C M and Im(f) C N to be the kernel and image of f considered as an abelian group homomorphism. (1.4) Definition. (1) Let F be a field. Then an Fmodule V is called a vector space over F. (2) If V and W are vector spaces over the field F then a linear transfor
mation from V to W is an Fmodule homomorphism from V to W. (1.5) Examples. (1)
Let G be any abelian group and let g E G. If n E Z then define the scalar multiplication ng by
(n terms) if n > 0,
ng=
0
(g) +
ifn=0, + (g) (n terms) ifn < 0.
Using this scalar multiplication G is a Zmodule. Furthermore, if G and H are abelian groups and f : G a H is a group homomorphism, then f is also a Zmodule homomorphism since (if n > 0) f (ng) = f (9 + ... + 9) = f (9) + ... + f(9) = nf(9) and
f(g) = f(g).
Chapter 3. Modules and Vector Spaces
110
(2) Let R be an arbitrary ring. Then R" is both a left and a right Rmodule via the scalar multiplications
a(bl, ... b") = (abi, ... ,ab") and
(bi, ...
, b" )a
= (bia, ... , b"a).
(3) Let R be an arbitrary ring. Then the set of matrices M,,,,"(R) is both a left and a right Rmodule via left and right scalar multiplication of matrices, i.e., ent;j (aA) = a ent,J(A) and
ent,3(Aa) = (ent,j(A))a.
(4) As a generalization of the above example, the matrix multiplication maps
Mm(R) x M,,,,"(R)
Mm."(R)
(A, B) . AB and
Mm,"(R) x M"(R)
Mm,"(R)
(A, B)  AB make M,,,,"(R) into a left Mm(R)module and a right M"(R)module. (5) If R is a ring then a left ideal I C R is a left Rmodule, while a right
ideal J C R is a right Rmodule. In both cases the scalar multiplication is just the multiplication of the ring R. (6) If R is a ring and I C R is an ideal then the quotient ring R/I is both a left Rmodule and a right Rmodule via the multiplication maps
(a, b + I)
ab + I
and
R/IxR+R/I (a+I,b).sab+l. (7)
M is defined to be an Ralgebra if M is both an Rmodule and a ring, with the ring addition being the same as the module addition, and the multiplication on M and the scalar multiplication by R satisfying the following identity: For every r E R, m1, m2 E M,
(1.1)
r(mlm2) = (rmi)m2 = ml(rm2).
3.1 Definitions and Examples
111
For example, every ring is a Zalgebra, and if R is a commutative ring,
then R is an Ralgebra. Let R and S be rings and let 4 : R + S be a ring homomorphism with Im(m) C C(S) = (a E S : ab = ba for all b E S), the center of S. If M is an Smodule, then M is also an Rmodule using the scalar multiplication am = (4(a))m for all a E R and m E M. Since S itself is an Smodule, it follows that S is an Rmodule, and moreover, since Im(¢) C C(S), we conclude that S is an Ralgebra. As particular cases of this construction, if R is a commutative ring, then the polynomial ring R[X] and the matrix ring
M are Rmodules then HomR(M, N) is an abelian group via the operation (f + g)(m) = f (m) + g(m). However, if we try to make HomR(M, N) into an Rmodule in the natural way by defining of by the formula (a f)(m) = a(f (m)) we find that the function a f need not be an Rmodule homomorphism unless R is a commutative ring. To see this, note that (af)(rm) = a(f (rm)) = a(r(f (m))) = ar f (m).
This last expression is equal to r(a f) (m) = ra f (m) if R is a commutative ring, but not necessarily otherwise. Thus, if R is a commutative ring, then we may consider HomR(M, N) as an Rmodule for all M, N, while if R is not commutative then HomR(M, N) is only an abelian group. Since EndR(M) is also a ring using composition of Rmodule homomorphisms as the multiplication, and since there is a ring homomorphism 0 : R  EndR(M) defined by O(a) = a 1M where IM denotes the identity homomorphism of M, it follows from Example 1.5 (7) that EndR(M) is an Ralgebra if R is a commutative ring. (9) If G is an abelian group, then Homz(Z, G) G. To see this, define
$ : Homz(Z, G)  G by 4i(f) = f (1). We leave it as an exercise to check that 4) is an isomorphism of Zmodules. (10) Generalizing Example 1.5 (9), if M is an Rmodule then
HomR(R,M)^='M as Zmodules via the map 4) : HomR(R, M) + M where 0(f) = f (1).
(11) Let R be a commutative ring, let M be an Rmodule, and let S C EndR(M) be a subring. (Recall from Example 1.5 (8) that EndR(M) is a ring, in fact, an R algebra.) Then M is an Smodule by means of the scalar multiplication map S x M M defined by (f, m) '' f (m). (12) As an important special case of Example 1.5 (11), let T E EndR(M) and define a ring homomorphism 0 : R[X] EndR(M) by sending X to T and a E R to alM. (See the polynomial substitution theorem (Theorem 2.4.1).) Thus, if
f (X) =ao+a1X
+...+aX"
Chapter 3. Modules and Vector Spaces
112 then
O(f(X)) We will denote O(f (X)) by the symbol f (T) and we let Im(O) = R[T]. That is, R[T] is the subring of EndR(M) consisting of "polynomials" in T. Then M is an R[TJ module by means of the multiplication
f (T)m = f (T)(m).
Using the homomorphism 0: R(X) , R[T] we see that M is an R[X]module using the scalar multiplication
f (X)m = f (T)(m). This example is an extremely important one. It provides the basis for applying the theory of modules over principal ideal domains to the study of linear transformations; it will be developed fully in Section 4.4.
(13) We will present a concrete example of the situation presented in Ex
ample 1.5 (12). Let F be a field and define a linear transformation
T : F2 + F2 by T(ul, u2) = (u2, 0). Then T2 = 0, so if f (X) = + amXm E F[XJ, it follows that f (T) = ao1Fa + a1T. ao + a1X + Therefore the scalar multiplication f (X )u for u E F2 is given by f(X) . (U1, U2) = f(T)(u1,u2) = (ao1F2 +a,T)(u1,u2) = (aoui +a1u2,aou2)
3.2 Submodules and Quotient Modules Let R be a ring and M an Rmodule. A subset N C M is said to be a submodule (or Rsubmodule) of M if N is a subgroup of the additive group of M that is also an Rmodule using the scalar multiplication on M. What this means, of course, is that N is a submodule of M if it is a subgroup of M that is closed under scalar multiplication. These conditions can be expressed as follows.
(2.1) Lemma. If M is an Rmodule and N is a nonempty subset of M, then N is an Rsubmodule of M if and only if amt + bm2 E N for all
m1, m2 N and a, bE R. Proof. Exercise.
3.2 Submodules and Quotient Modules
113
If F is a field and V is a vector space over F, then an Fsubmodule of V is called a linear subspace of V. (2.2) Examples. If R is any ring then the Rsubmodules of the Rmodule R are precisely the left ideals of the ring R. If G is any abelian group then G is a Zmodule and the Zsubmodules of G are just the subgroups of G.
Let f : M  N be an Rmodule homomorphism. Then Ker(f) C M and Im(f) C N are Rsubmodules (exercise).
Continuing with Example 1.5 (12), let V be a vector space over a field F and let T E EndF(V) be a fixed linear transformation. Let VT denote V with the FIX)module structure determined by the linear transformation T. Then a subset W C V is an F[X]submodule of the module VT if and only if W is a linear subspace of V and T (W) C W, i.e., W must be a Tinvariant subspace of V. To see this, note that
X w = T(w), and if a E F, then a w = awthat is to say, the action of the constant polynomial a E F[X] on V is just ordinary scalar multiplication, while the action of the polynomial X on V is the action of T on V. Thus, an F[X]submodule of VT must be a Tinvariant subspace of V. Conversely, if W is a linear subspace of V
such that T(W) C W then Tm(W) C W for all m > 1. Hence, if f (X) E F[X] and W E W then f (X) w = f (T)(w) E W so that W is closed under scalar multiplication and thus W is an F[X]submodule of V.
(2.3) Lemma. Let M be an Rmodule and let (NQ}QEA be a family of submodules of M. Then N Z n0EA N. is a submodule of M. Proof. Exercise.
0
We now consider quotient modules and the noether isomorphism theorems. Let M be an Rmodule and let N C_ M be a submodule. Then N is a subgroup of the abelian group M, so we can form the quotient group M/N. Define a scalar multiplication map on the abelian group M/N by
a(m + N) = am + N for all a E R, m + N E M/N. Since N is an Rsubmodule of M, this map is well defined. Indeed, if m + N = m' + N then
mm' E N so that amam'= a(mm') E N so that am+N = am'+N. The resulting Rmodule M/N is called the quotient module of M with respect to the submodule N. The noether isomorphism theorems, which we have seen previously for groups and rings, then have direct analogues for Rmodules.
(2.4) Theorem. (First isomorphism theorem) Let M and N be modules over the ring R and let f : M N be an Rmodule homomorphism. Then Im(f).
114
Chapter 3. Modules and Vector Spaces
Proof. Let K = Ker(f). From Theorem 1.3.10 we know that f : M/K Im(f) defined by f (m+K) = f (m) is a welldefined isomorphism of abelian groups. It only remains to check that f is an Rmodule homomorphism. But
f(a(m+K)) = f(am+K) = f(am) = af(m) = af(m+K) for all m E M and a E R, so we are done. 0 (2.5) Theorem. (Second isomorphism theorem) Let M be an Rmodule and let N and P be submodules. Then there is an isomorphism of Rmodules
(N+P)/PAN/(NnP). Proof. Let rr : M , M/P be the natural projection map and let rro be the restriction of rr to N. Then ao is an Rmodule homomorphism with Ker(rro) = N n P and Im(iro) = (N + P)/P. The result then follows from the first isomorphism theorem. 0
(2.6) Theorem. (Third isomorphism theorem) Let M be an Rmodule and let N and P be submodules of M with P C N. Then
111/N = (M/P)/(N/P). Proof. Define f : Al/P . MIN by f (m+ P) = m+N. This is a welldefined Rmodule homomorphism and
Ker(f)={m +P:m+N=N}={m +P:mEN}=NIP. The result then follows from the first isomorphism theorem (Theorem 2.4).
0 (2.7) Theorem. (Correspondence theorem) Let Al be an Rmodule, N a submodule, and A : M  MIN the natural projection map. Then the function P  P/N defines a onetoone correspondence between the set of all submodules of M that contain N and the set of all submodules of M/N. Proof. Exercise.
0
(2.8) Definition. If S is a subset of an Rmodule M then (S) will denote the intersection of all the submodules of M that contain S. This is called the submodule of M generated by S, while the elements of S are called generators of (S). Thus, (S) is a submodule of M that contains S and it is contained in every submodule of M that contains S, i.e., (S) is the smallest submodule of M containing S. If S = {x1, ... ,xn} we will usually write (xi, ... ,xn)
3.2 Submodules and Quotient Modules
115
rather than ({x1, ... , for the submodule generated by S. There is the following simple description of (S).
(2.9) Lemma. Let M be an Rmodule and let S C M. If S = 0 then (S) =
{0},while (S)={E 1a;si:nEN,a;ER, s,ES,1jE J f j (x j ). Note that this sum can be considered as a welldefined finite sum since xj = 0 except for finitely many indices j E J. (Note that this construction does not work for infinite direct products.) We leave it as an exercise to check that f is an Rmodule homomorphism.
The characterization of when an Rmodule M is isomorphic to the direct sum of submodules is essentially the same as the characterization provided in Theorem 3.1. We state the result, but the verification is left as an exercise.
(3.2) Theorem. Let M be an Rmodule and let {Mj}jEJ be a family of submodules such that ,E.,Mj = (UjEJ Mj), and (1) M = >F,EJ Mj = {0} for every k E J. (2) Mk n
3.3 Direct Sums, Exact Sequences, and Hom
Then
121
MMi. jEJ
0
Proof. Exercise.
(3.3) Definition. If M is an Rmodule and MI C M is a submodule, we say that MI is a direct summand of M, or is complemented in M, if there is a submodule M 2 C M such that M ? 5 MI ®M2.
(3.4) Example. Let R = Z and M = Z e. If MI = (p) then MI is not complemented since MI is the only subgroup of M of order p, so condition (2) of Theorem 3.1 is impossible to satisfy.
The concept of exact sequences of Rmodules and Rmodule homomorphisms and their relation to direct summands is a useful tool to have available in the study of modules. We start by defining exact sequences of Rmodules.
(3.5) Definition. Let R be a ring. A sequence of Rmodules and Rmodule homomorphisms
is said to be exact at Mi if Im(fi) = Ker(fi+I ). The sequence is said to be exact if it is exact at each Mi. As particular cases of this definition note that
(1) 0 + MI f+ M is exact if and only if f is injective,
(2) M + M2  0 is exact if and only if g is surjective, and (3) the sequence 0  . MI
(3.1)
_ M 9+ M2'0
is exact if and only if f is injective, g is surjective, and Im(f) = Ker(g). Note that the first isomorphism theorem (Theorem 2.4) then shows
that M2 ?t M/ Im(f). M/ Im(f) is called the cokernel of f and it is denoted Coker(f). (3.6) Definition. (1) The sequence (3.1), if exact, is said to be a short exact sequence. (2) The sequence (3.1) is said to be a split exact sequence (or just split)
if it is exact and if Im(f) = Ker(g) is a direct summand of M.
Chapter 3. Modules and Vector Spaces
122
In the language of exact sequences, Proposition 2.12 can be stated as follows:
(3.7) Proposition. Let 0 . M1 + M i M2  0 be a short exact sequence of Rmodules. If Ml and M2 are finitely generated, then so is M, and moreover, µ(M) (1). A short exact sequence
0  MI
M 'p s P + 0,
in particular, includes a surjection V : M i P. Now take N = P in part (4). Thus,
ii.: HomR(P, M) + HomR(P, P) is surjective. Choose /3 : P + M with t/i.(,0) = lp. Then /3 splits the short exact sequence and the result is proved. 0
Chapter 3. Modules and Vector Spaces
138
(5.2) Definition. An Rmodule P satisfying any of the equivalent conditions of Theorem 5.1 is called projective. As noted before Theorem 5.1, projective modules are introduced as the class of modules possessing the property that free modules were shown to possess in Proposition 4.16. Therefore, we have the following fact:
(5.3) Proposition. Free Rmodules are projective. Proof.
(5.4) Corollary. Let R be an integral domain. If P is a projective Rmodule, then P is torsionfree.
Proof. By Theorem 5.1 (2), P is a submodule of a free module F over R. According to Proposition 4.8, every free module over an integral domain is torsionfree, and every submodule of a torsionfree module is torsionfree. (5.5) Corollary. An Rmodule P is a finitely generated projective Rmodule if and only if P is a direct summand of a finitely generated free Rmodule.
Proof. Suppose that P is finitely generated and projective. By Proposition 4.14, there is a free presentation
0 K  F , P  0 such that F is free and p(F) = µ(P) < oo. By Theorem 5.1, P is a direct summand of F. Conversely, assume that P is a direct summand of a finitely generated free Rmodule F. Then P is projective, and moreover, if P ® P  F then
F/P  P so that P is finitely generated. (5.6) Examples. (1) (2)
(3)
Every free module is projective.
Suppose that m and n are relatively prime natural numbers. Then as abelian groups Z,,,,, ?5 Z,,, ® Z,,. It is easy to check that this isomorphism is also an isomorphism of Z,,,,,modules. Therefore, Z,,, is a direct summand of a free Z,,,,,module, and hence it is a projective Z,,,,,module. However, Z,,, is not a free Z,,,,, module since it has fewer than mn elements. Example 5.6 (2) shows that projective modules need not be free. We will present another example of this phenomenon in which the ring R is an integral domain so that simple cardinality arguments do not suffice.
Let R = Z(v5] and let I be the ideal I = (2, 1 +
= (al, a2). It
is easily shown that I is not a principal ideal, and hence by Example 4.6 (6), we see that I cannot be free as an Rmodule. We claim that I
3.5 Projective Modules
139
E R, let F be a is a projective Rmodule. To see this, let b = 1 free Rmodule with basis Is,, 82}, and let 0: F  I be the Rmodule homomorphism defined by c(rjs1 + r2s2) = r1al + r2a2.
Now define an Rmodule homomorphism a : 1 + F by
a(a) = as, + ((ab)/2)s2. Note that this makes sense because 2 divides ab for every a E I. Now for a E I, 0 o a(a) = b(asl + ((ab)/2)s2)
= aal + ((ab)/2)a2
= aal + aa2b/2 = 2a + 3a =a so that a is a splitting of the surjective map 0. Hence, F and by Theorem 5.1, I is a projective Rmodule.
Ker(O) ® I
Concerning the construction of new projective modules from old ones, there are the following two simple facts:
(5.7) Proposition. Let { Pj }jEJ be a family of Rmodules, and let P = ®jEJ Pi. Then P is projective if and only if Pj is projective for each j E J.
Proof. Suppose that P is projective. Then by Theorem 5.1, there is an Rmodule P' such that P ® P = F is a free Rmodule. Then
F=P®P'=(®Pj)ED P', jEJ
and hence, each Pj is also a direct summand of the free Rmodule F. Thus, Pj is projective. Conversely, suppose that Pj is projective for every j E J and let P,' be an Rmodule such that Pj ® Pj' = F, is free. Then
P®((DPj')(P,®P;) jEJ
jEJ
® Fj. jEJ
Since the direct sum of free modules is free (Example 4.6 (8)), it follows that P is a direct summand of a free module, and hence P is projective.
140
Chapter 3. Modules and Vector Spaces
(5.8) Proposition. Let R be a commutative ring and let P and Q be finitely generated projective Rmodules. Then HomR(P, Q) is a finitely generated projective Rmodule.
Proof. Since P and Q are finitely generated projective Rmodules, there are Rmodules P and Q' such that P®P' and Q®Q' are finitely generated free modules. Therefore, by Theorem 4.11, HomR(P ® P, Q (D Q') is a finitely generated free Rmodule. But HomR(P (D P', Q ® Q') ?' HomR(P, Q) ® HomR(P, Q') ® HomR(P', Q) ® HomR(P, Q')
so that HomR(P, Q) is a direct summand of a finitely generated free Rmodule, and therefore, it is projective and finitely generated by Corollary
0
5.5.
Example 5.6 (3) was an example of an ideal in a ring R that was projective as an Rmodule, but not free. According to Example 4.6 (6), an ideal I in a ring R is free as an Rmodule if and only if the ideal is principal.
It is a natural question to ask which ideals in a ring R are projective as Rmodules. Since this turns out to be an important question in number theory, we will conclude our brief introduction to the theory of projective modules by answering this question for integral domains R.
(5.9) Definition. Let R be an integral domain and let K be the quotient field of R. An ideal I C R is said to be invertible if there are elements a1, ... , an E I and bl, ... , bn E K such that (5.1) (5.2)
b=I C_ R for 1 < i < n, and albs + + anbn = 1.
(5.10) Examples. (1)
If I C R is the principal ideal I = (a) where a 0 0, then I is an invertible ideal. Indeed, let b = 1/a E K. Then any x E I is divisible by a in R so that bx = (1/a)x E R, while a(l/a) = 1.
(2) Let R = Z[vr5] and let I = (2, 1 + ). Then it is easily checked that I is not principal, but I is an invertible ideal. To see this, let a1 =2,a2=1+,,/5,b1 =1, and b2 = (1  ,/5)/2. Then albs + a2b2 = 2 + 3 = 1.
Furthermore, a1b2 and a2b2 are in R, so it follows that b2I C R, and we conclude that I is an invertible ideal. The following result characterizes which ideals in an integral domain
R are projective modules. Note that the theorem is a generalization of Example 5.6 (3):
3.5 Projective Modules
141
(5.11) Theorem. Let R be an integral domain and let I C R be an ideal. Then I is a projective Rmodule if and only if I is an invertible ideal. Proof. Suppose that 1 is invertible and choose al, ... , an E I and b1, ... , bn in the quotient field K of R so that Equations (5.1) and (5.2) are satisfied.
Let 0: Rn ' I be defined by
O(xi, ... , xn) = alxl + ... + anxn, and define 6 : I + Rn by /3(a) = (abl, ... , abn).
Note that abi E R for all i by Equation (5.1). Equation (5.2) shows that n
O o /3(a) = E ai(abi) = a t aibi =a i_1
i=1
for every a E I. Therefore 0 o /j = 1 p and Theorem 3.9 implies that I is a direct summand of the free Rmodule Rn, so I is a projective Rmodule. Conversely, assume that the ideal 19 R is projective as an Rmodule. Then I is a direct summand of a free Rmodule F, so there are Rmodule
homomorphisms 0 : F + I and 3 : I ' F such that 0 o /3 = 11. Let S = {xj}jEJ be a basis of F. Given x E 1, /3(x) E F can be written uniquely as
cjxj.
/3(x) _
(5.3)
jEJ
For each j E J, let tlj(x) = cj. This gives a function Oj : I R, which is easily checked to be an Rmodule homomorphism. If aj = O(xj) E I, note that for each x E I, Oj(x) = 0 except for at most finitely many j E J; for each x E I, Equation (5.3) shows that
(5.4) (5.5)
x = 0(0(x)) = 1: 'pj(x)aj jEJ
Given x j4 0 E I and j E J, define b, E K (K is the quotient field of R) by (5.6)
bj = Oj(x)
X
The element bj E K depends on j E J but not on the element x 54 0 E I. To see this, suppose that x' 54 0 E I is another element of I. Then X'pj(x) = Oj(x'x) =''j(xx) = x1pj(x )
142
Chapter 3. Modules and Vector Spaces
so that >' 1(x)/x = 1(ii(x')/x'. Therefore, for each j E J we get a uniquely defined bi E K. By property (5.4), at most finitely many of the bi are not 0. Label the nonzero bi by b1, ... , bn. By property (5.5), if x $ 0 E I then n
n
x= E V)i(x)a.i =>(bix)ai =x .i=1
.i=1
n
(bia)). i=1
Cancelling x $ 0 from this equation gives a1b1 + ... + anbn = 1
where a1, ... , an E I and b1 , bn E K. It remains to check that b,I C R for 1 < j < n. But if x $ 0 E I then bi = tki(x)/x so that bix = V)j(x) E R. Therefore, I is an invertible ideal and the theorem is proved. 0 (5.12) Remark. Integral domains in which every ideal is invertible are known
as Dedekind domains, and they are important in number theory. For example, the ring of integers in any algebraic number field is a Dedekind domain.
3.6 Free Modules over a PID In this section we will continue the study of free modules started in Section 3.4, with special emphasis upon theorems relating to conditions which ensure that a module over a PID R is free. As examples of the types of theorems to be considered, we will prove that all submodules of a free Rmodule are free and all finitely generated torsionfree Rmodules are free, provided that the ring R is a PID. Both of these results are false without the assumption that R is a PID, as one can see very easily by considering an integral domain R that is not a PID, e.g., R = Z[X], and an ideal I C R that is not principal, e.g., (2, X) C Z[XJ. Then I is a torsionfree submodule of R that is not free (see Example 4.6 (6)). Our analysis of free modules over PIDs will also include an analysis of which elements in a free module M can be included in a basis and a criterion
for when a linearly independent subset can be included in a basis. Again, these are basic results in the theory of finitedimensional vector spaces, but the case of free modules over a PID provides extra subtleties that must be carefully analyzed. We will conclude our treatment of free modules over PIDs with a fundamental result known as the invariant factor theorem for finite rank submodules of free modules over a PID R. This result is a farreaching generalization of the freeness of submodules of free modules, and it is the basis
3.6 Free Modules over a PID
143
for the fundamental structure theorem for finitely generated modules over PIDs which will be developed in Section 3.7. We start with the following definition:
(6.1) Definition. Let M be a free Rmodule. Then the free rank of M, denoted freerankR(M), is the minimal cardinality of a basis of M.
Since we will not be concerned with the fine points of cardinal arithmetic, we shall not distinguish among infinite cardinals so that freerankR(M) E Z+ U {oo}.
Since a basis is a generating set of M, we have the inequality u(M) < freerankR(M). We will see in Corollary 6.18 that for an arbitrary commutative ring R and for every free Rmodule, freerankR(M) = p(M) and all bases of M have this cardinality.
(6.2) Theorem. Let R be a PID, and let M be a free Rmodule. If N C M is a submodule, then N is a free Rmodule, and freerankR(N) < freerankR(M). Proof. We will first present a proof for the case where freerankR(M) < 00. This case will then be used in the proof of the general case. For those who are only interested in the case of finitely generated modules, the proof of the second case can be safely omitted.
Case 1. freerankR(M) < 00.
We will argue by induction on k = freerankR(M). If k = 0 then M = (0) so N = (0) is free of freerank 0. If k = 1, then M is cyclic so M = (x) for some nonzero x E M. If N = (0) we are done. Otherwise, let
I = {a ER: ax EN). Since! is an ideal ofRandRisaPID,I=(d); since N j4 (0), d 54 0. If Y E N then y = ax = rdx E (dx) so that N = (dx) is a free cyclic Rmodule. Thus freerankR(N) = 1 and the result is true for k = 1. Assume by induction that the result is true for all M with freerank k, and let M be a module with freerankR(M) = k+l. Let S = {x1, ... xk+l } be a basis of M and let Mk = (xl, ... ,xk). If N C Mk we are done by induction. Otherwise N n Mk is a submodule of Mk which, by induction, is free of freerank e < k. Let {yl, ... , ye} be a basis of N n Mk. By Theorem 2.5
N/(N n Mk) 25 (N + Mk)/Mk C M/Mk = (xk+I + Mk)By the k = 1 case of the theorem, (N + Mk)/Mk is a free cyclic submodule of M/Mk with basis dxk+1 + Mk where d 54 0. Choose ye+l E N so that ye+i = dxk+i + x' for some x' E Mk. Then (N + Mk) lMk = (ye+i + Mk).
144
Chapter 3. Modules and Vector Spaces
We claim that S' = {yl, ... , ye, yt+1 } is a basis of N. To see this, let y E N. Then y + Mk = ae+i(ye+i + Mk) so that y  at+lyt+1 E N n Mk, which implies that y  at+Iyt+1 = a1y1 + aeye. Thus S' generates N. Suppose
that aly, + +ae+lyt+1 = 0. Then at+1(dxk+1 +x')+aIy1 + +aeye = 0 so that ae+ldxk+l E Mk. But S is a basis of M so we must have ae+ld = 0; since d 34 0 this forces at+1 = 0. Thus a1y1 + + aeye = 0 which implies that a1 = = at = 0 since {y1, . . , yt} is linearly independent. Therefore S' is linearly independent and hence a basis of N, so that N is free with freerankR(N) < e + 1 < k + I. This proves the theorem in Case 1. Case 2. freerankn(M) = oo.
Since (0) is free with basis 0, we may assume that N # (0). Let S =
{xj}jEJ be a basis of M. For any subset K C J let MK = ({xk}kEK) and let NK = N n MK. Let T be the set of all triples (K, K', f) where K' C K C J and f : K' + NK is a function such that (f (k)}kEK' is a basis of NK We claim that T # 0. .
Since N $ (0) there is an x j4 0 E N, so we may write x = aixj, + + }. But MK is a free Rmodule with freerankR(MK) < k < oo and NK is a nonzero submodule. By Case 1, NK is free with freerankR(NK) = e < k. Let {y1, ... yt} be a basis of NK, and let K' = (j 1, ... , jt}, and define f : K' NK by f (j;) = y, for 1 < i < e. Then (K, K', f) E T so that T A 0, as claimed. Now define a partial order on T by setting (K, K', f) < (L, L', g) if K C L, K' C L', and 9IK' = f. If {(KQ, KQ, fQ)}QEA C T is a chain, then (UQEA KQ, UQEA KQ, F) where FIK' = fQ is an upper bound in T for the chain. Therefore, Zorn's lemma applies and there is a maximal element akxjk. Hence X E NK where K = {{ j 1 ,,
(K, K', f) of T. Claim. K = J. Assuming the claim is true, it follows that MK = M, NK = N n MK = N, and {f(k)}kEK' is a basis of N. Thus, N is a free module (since it has a basis), and since S was an arbitrary basis of M, we conclude that N has a basis of cardinality < freerankR(M), which is what we wished to prove. It remains to verify the claim. Suppose that K j4 J and choose j E
J \ K. Let L = K U {j}. If NK = NL then (K, K', f)
(L, K', f),
contradicting the maximality of (K, K', f) in T. If NK 36 NL, then NL/(NL n MK)
(NL + MK)/MK C ML/MK = (x3 + MK)
By Case 1, (NL + 1MIK)/MK is a free cyclic submodule with basis dxj + MK
where d # 0. Choose z E NL so that z = dxj + w for some w E MK. Then (NL + MK)/MK = (z + MK). Now let L' = K' U {j} and define L' +NL by
f,(k) = J f (k) if k E K', Z
ifk=j.
3.6 Free Modules over a PID
145
We need to show that { f'(k)}kEL' is a basis of NL. But if x E NL then x + MK = cz + MK for some c E R. Thus x  cz E MK n N = NK so that
x  cz =
bkf(k) kE K'
where bk E R. Therefore, { f (k)}kEL' generates NL. Now suppose EkEL' bkf'(k) = 0. Then
b,z+1: bkf(k)=0 kE K'
so that
db,x,+b,w+ E bkf(k) =0. kEK'
That is, db,x, E MK n (x,) = (0), and since S = {xt}tEJ is a basis of M, we must have db, = 0. But d j4 0, so b, = 0. This implies that EkE K, bk f (k) = 0. But { f (k) }kE K' is a basis of NK, so we must have bk = 0 for all k E K'. Thus { f'(k)}kEL' is a basis of NL. We conclude that (K, K', f) (L, L', f'), which contradicts the maximality of (K, K', f). Therefore, the claim is verified, and the proof of the theorem is complete. (6.3) Corollary. Let R be a PID and let P be a projective Rmodule. Then P is free. Proof. By Proposition 4.14, P has a free presentation
0+K+F+P+0. P ® K. Since P is projective, this exact sequence splits and hence F Therefore, P is isomorphic to a submodule of F, and Theorem 6.2 then shows that P is free. (6.4) Corollary. Let M be a finitely generated module over the PID R and let N C M be a submodule. Then N is finitely generated and
µ(N) < µ(M). Proof. Let
0 K F 0+My0
be a free presentation of M such that freerank(F) = p(M) < oo, and let NI = 01(N). By Theorem 6.2, N1 is free with µ(N1) < freerank(NI) < freerank(F) = µ(M). Since N = O(NI ), we have µ(N) < µ(N1), and the result is proved.
146
Chapter 3. Modules and Vector Spaces
(6.5) Remark. The hypothesis that R be a PID in Theorem 6.2 and Corollaries 6.3 and 6.4 is crucial. For example, consider the ring R = Z[X] and
let M = R and N = (2, X). Then M is a free Rmodule and N is a submodule of M that is not free (Example 4.6 (6)). Moreover, R = Z[V/5],
P = (2, 1 + /) gives an example of a projective Rmodule P that is not free (Example 5.6 (3)). Also note that 2 = µ(N) > µ(M) = 1 and 2 = µ(P) > I = u(R). Recall that if M is a free module over an integral domain R, then M is torsionfree (Proposition 4.8). The converse of this statement is false even
under the restriction that R be a PID. As an example, consider the Zmodule Q. It is clear that Q is a torsionfree Zmodule, and it is a simple exercise to show that it is not free. There is, however, a converse if the module is assumed to be finitely generated (and the ring R is a PID).
(6.6) Theorem. If R is a PID and M is a finitely generated torsionfree Rmodule, then M is free and freerankR(M) = µ(M).
Proof. The proof is by induction on µ(M). If µ(M) = 1 then M is cyclic with generator {x}. Since M is torsionfree, Ann(x) = {0}, so the set {x} is linearly independent and, hence, is a basis of M.
Now suppose that µ(M) = k > 0 and assume that the result is true for all finitely generated torsionfree Rmodules M' with µ(M') < k. Let {x1, ... , xk } be a minimal generating set for M, and let
M1={xERI:axE(x1)
for some a0OER}.
Then M/M1 is generated by {x2 + M1, ... , xk + M1 } so that µ(M/Ml) _ j < k  1. If ax E M1 for some a 54 0 E R, then from the definition of M1, b(ax) E (xl) for some b 0 0. Hence x E M1 and we conclude that M/Ml is torsionfree. By the induction hypothesis, M/M1 is free of freerank j. Then Corollary 4.17 shows that ML, M1® (M/Ml ). We will show that M1 is free of freerank 1. It will then follow that
k=µ(l41)2jEJ(aj/d)xj), so if d is not a unit then x is not primitive. Conversely, if d = 1 and x = ay then
E ajxj = x jEJ = ay(
=
l
a(Ebjxj)
jEJ
_ E abjxj. jEJ
Since S = {xj}jEJ is a basis, it follows that aj = abj for all j E J. That is, a is a common divisor of the set {aj}jEJ so that a I d = 1. Hence a is a unit and x is primitive. (6.14) Lemma. Let R be a PID and let M be a finitely generated Rmodule.
If x E M has Ann(x) = (0), then we may write x = ax' where a E R and x' is primitive. (In particular, if M is not a torsion module, then M has a primitive element.) Proof. Let xo = x. If xo is primitive we are done. Otherwise, write xo = alxl
where al E R is not a unit. Then (xo) 5 (x1). To see this, it is certainly true that (xo) C (xi). If the two submodules are equal then we may write
xl = bxo so that xo = aixl = albxo, i.e., (1  alb) E Ann(xo) = (0). Therefore, 1 = alb and al is a unit, which contradicts the choice of al.
Now consider xl. If xl is primitive, we are done. Otherwise xl = a2x2 where a2 is not a unit, and as above we conclude that (xl) (x2). Continuing in this way we obtain a chain of submodules (6.1)
(xo) 5 (xi) 5 (x2) 5 ...
Either this chain stops at some i, which means that xi is primitive, or (6.1) is an infinite properly ascending chain of submodules of M. We claim that the latter possibility cannot occur. To see this, let N = Ui_° I (xi). Then N is a submodule of the finitely generated module M over the PID R so that N is also finitely generated by {yl,... , yk } (Corollary 6.4). Since (xo) C (xl) C , there is an i such that {yi,...,yk} C (xi). Thus N = (xi) and hence (x,) = (xi+i) = , which contradicts having an infinite properly
150
Chapter 3. Modules and Vector Spaces
ascending chain. Therefore, xi is primitive for some i, and if we let x' = xi we conclude that x = ax' where a = ala2 . ai. 0 (6.15) Remark. Suppose that M is a free Rmodule, where R is a PID, and x E M. Then Ann(x) = (0), so x = ax' where x' is a primitive element of
M. If S = {xj }jEJ is a basis of M, then we may write x' = E,E J bjxj so that x = ax=Eabjxj= cjxj. jEJ
jEJ
Since gcd({bj}jEJ) = 1 (by Lemma 6.13) we see that a = gcd({cj}jEJ). The element a E R, which is uniquely determined by x up to multiplication by a unit of R, is called the content of x E M and is denoted c(x). (Compare with the concept of content of polynomials (Definition 2.6.3).) Thus, any x E M can be written
x = c(x) i
(6.2)
where x' is primitive.
(6.16) Theorem. Let R be a PID and let M be a fee Rmodule with
rank(M) = k = µ(M) = freerank(M). If x E M is primitive, then M has a basis of k elements containing x.
Proof. Assume first that k < oo and proceed by induction on k. Suppose k = 1 and let M have a basis {x1 }. Then x = ax1 for some a E R. Since x is primitive, it follows that a is a unit so that (x) = (x1) = M, hence {x} is a basis of M. The case k = 2 will be needed in the general induction step, so we present it separately. Thus suppose that M has a basis {x1, x2} and let x = rx1 + sx2 where r, s E R. Since x is primitive, gcd{r, s} = 1, so we may write ru + sv = 1. Let x2 = vxl + ux2. Then
x1 =uxsx2 and
x2 = vx + rx2.
Hence, (x, x2) = M. It remains to show that {x, x2} is linearly independent. Suppose that ax + bx2 = 0. Then a(rx1 + sx2) + b(vx1 + UX2) = 0. Since {x1i x2} is a basis of M, it follows that
arbv=0 and
3.6 Free Modules over a PID
151
as+bu=0. Multiplying the first equation by u, multiplying the second by v, and adding shows that a = 0, while multiplying the first by s, multiplying the second by r, and adding shows that b = 0. Hence, {x, x2} is linearly independent and, therefore, a basis of M. Now suppose that u(M) = k > 2 and that the result is true for all free Rmodules of rank < k. By Theorem 6.6 there is a basis {x1, ... , xk } of M.
Letx=Ekla;x;.Ifak=0then xEM1=(x1,...,xk_1),sobyinduction there is a basis {x, x2, ... ,xk_1} of M1. Then {x,x2, ... ,xk_1, xk} is a basis of M containing x. Now suppose that ak # 0 and let y = Ei=1 a;x;. If y = 0 then x = akxk, and since x is primitive, it follows that ak is a unit
of R and {x1, ... ,xk_1, x} is a basis of M containing x in this case. If 0 then there is a primitive y' such that y = by' for some b E R. In y particular, y' E M1 so that M1 has a basis {y', x2, ... , x'ti11 and hence M has a basis {y', x2, ... , xk_1, xk}. But x =akxk + y = akxk + by' and gcd(ak, b) = 1 since x is primitive. By the previous case (k = 2) we conclude
that the submodule (xk, y') has a basis {x, y"}. Therefore, M has a basis {x, x2, ... , xk_1, y"} and the argument is complete when k = µ(M) < oo. If k = oo let {x, }jE J be a basis of M and let x = 1 aixj, for
some finite subset I = 01, ... , j } c J. If N = (xj...... xi,,) then x is a primitive element in the finitely generated module N, so the previous argument applies to show that there is a basis {x, x2, ... , x' } of N. Then {x, x2, ... ,xn} U {x,}iEJ\I is a basis of M containing x. 0 (6.17) Corollary. If M is a free module over a PID R, then every basis of M contains µ(M) elements. Proof. In case µ(M) < oo, the proof is by induction on µ(M). If µ(M) = 1 then M = (x). If {x1, x2} C M then x1 = a1x and and x2 = a2x so that a2x1 a, X2 = 0, and we conclude that no subset of M with more than one element is linearly independent. Now suppose that p(M) = k > 1 and assume the result is true for all free Rmodules N with µ(N) < k. Let S = {x) }JE J c M be any basis of M and choose x E S. Since x is primitive (being an element of a basis), Theorem 6.16 applies to give a basis {x, y2i ... , yk } of M with precisely µ(M) = k elements. Let N = M/(x) and let it : M  N be the projection map. It is clear that N is a free Rmodule with basis 7r(S) \ {ar(x)}. By Proposition 2.12 it follows that µ(N) > k 1, and since {7r(y2), ... , ir(yk)} generates N, we conclude that µ(N) = k  1. By induction, it follows that ISI  1 < oo and ISO  1 = k  1, i.e., ISI = k, and the proof is complete in case µ(M) < oo. In case µ(M) = oo, we are claiming that no basis of M can contain a finite number k E Z+ of elements. This is proved by induction on k, the proof being similar to the case µ(M) finite, which we have just done. We leave the details to the reader. 0
152
Chapter 3. Modules and Vector Spaces
(6.18) Corollary. Let R be any commutative ring with identity and let M be a free Rmodule. Then every basis of M contains µ(M) elements.
Proof. Let I be any maximal ideal of R (recall that maximal ideals exist by Theorem 2.2.16). Since R is commutative, the quotient ring R/I = K is a field (Theorem 2.2.18), and hence it is a P1D. By Proposition 4.13, the quotient module M/IM is a finitely generated free Kmodule so that Corollary 6.17 applies to show that every basis of M/IM has µ(M/IM) elements. Let S = {xj}JEJ be an arbitrary basis of the free Rmodule M and let it : M  M/IM be the projection map. According to Proposition 4.13, the set 7r(S) = {7r(xJ)}3EJ is a basis of M/IM over K, and therefore,
14(m)< V1 = µ(M/IM) 5 AM. Thus, µ(M) = IJI, and the corollary is proved.
0
(6.19) Remarks. (1) If M is a free Rmodule over a commutative ring R, then we have
proved that freerank(M) = µ(M) = the number of elements in any basis of M. This common number we shall refer to simply as the rank of M, denoted rankR(M) or rank(M) if the ring R is implicit. If R is a field we shall sometimes write dimR(M) (the dimension of M over
R) in place of rankR(M). Thus, a vector space M (over R) is finite dimensional if and only if dimR(M) = rankR(M) < oo. (2) Corollary 6.18 is the invariance of rank theorem for finitely generated free modules over an arbitrary commutative ring R. The invariance of rank theorem is not valid for an arbitrary (possibly noncommutative) ring R. As an example, consider the Zmodule M = ®nENZ, which is the direct sum of countably many copies of Z. It is simple to check
that M M ® M. Thus, if we define R = Endz(M), then R is a noncommutative ring, and Corollary 3.13 shows that
R = Endz(M) = Homz(M, M) Homz(M, M (D M) Homz(M, M) ® Homz(M, M)
R®R. The isomorphisms are isomorphisms of Zmodules. We leave it as an exercise to check that the isomorphisms are also isomorphisms of R
modules, so that R a, R2, and hence, the invariance of rank does not hold for the ring R. There is, however, one important class of noncommutative rings for which the invariance of rank theorem holds, namely, division rings. This will be proved in Proposition 7.1.14.
3.6 Free Modules over a PID
153
(6.20) Corollary. If M and N are free modules over a PID R, at least one of which is finitely generated, then M N if and only if rank(M) = rank(N).
Proof. If M and N are isomorphic, then p(M) = µ(N) so that rank(M) = rank(N). Conversely, if rank(M) = rank(N), then Proposition 4.9 gives a homomorphism f : M  N, which takes a basis of M to a basis of N. It is easy to see that f must be an isomorphism.
(6.21) Remark. One of the standard results concerning bases of finitedimensional vector spaces is the statement that a subset S = {x1, ... , x") of a vector space V of dimension n is a basis provided that S is either a spanning set or linearly independent. Half of this result is valid in the current context of finitely generated free modules over a PID. The set (2) C Z is linearly independent, but it is not a basis of the rank 1 Zmodule Z. There is, however, the following result.
(6.22) Proposition. Let M be a finitely generated free Rmodule of rank = k
where R is a PID. If S = {x1, ... ,xk} generates M, then S is a basis. Proof. Let T = {e1}?1 be the standard basis of Rk. Then there is a homo
morphism 0: Rk  M determined by 4,(e,) = xj. Since (S) = M, there is a short exact sequence
RkM + O where K = Ker(¢). Since M is free, Corollary 4.16 gives Rk  M ®K, and according to Theorem 6.2, K is also free of finite rank. Therefore,
k = rank(M) + rank(K) = k + rank(K) and we conclude that rank(K) = 0. Hence 0 is an isomorphism and S is a basis.
We will conclude this section with a substantial generalization of Theorem 6.2. This result is the crucial result needed for the structure theorem for finitely generated modules over a PID.
(6.23) Theorem. (Invariant factor theorem for submodules) Let R be a PID, let M be a free Rmodule, and let N C M be a submodule (which is automatically free by Theorem 6.2) of rank n < oo. Then there is a basis S of M, a subset {x1, ... , xn} C S, and nonzero elements ,, ..., sn E R such that (6.3)
{s1x1,
...
,snx,a}
is a basis of N
and (6.4)
si I si+1
for
1 < i < n  1.
154
Chapter 3. Modules and Vector Spaces
Proof. If N = (0), there is nothing to prove, so we may assume that N 96 (0)
and proceed by induction on n = rank(N). If n = 1, then N = (y) and {y} is a basis of N. By Lemma 6.14, we may write y = c(y)x where x E M is a primitive element and c(y) E R is the content of y. By Theorem 6.16, there is a basis S of M containing the primitive element x. If we let x1 = x and Si = c(y), then slxl = y is a basis of N, so condition (6.3) is satisfied; (6.4) is vacuous for n = 1. Therefore, the theorem is proved for n = 1. Now assume that n > 1. By Lemma 6.14, each y E N can be written as y = c(y) y' where c(y) E R is the content of y (Remark 6.15) and y' E M is primitive. Let S = {(c(y)) : y E N}. This is a nonempty collection of ideals of R. Since R is Noetherian, Proposition 2.5.10 implies that there is a maximal element of S. Let (c(y)) be such a maximal element. Thus, y E N and y = c(y) x, where x E M is primitive. Let s1 = c(y). Choose any basis T of M that contains x. This is possible by Theorem 6.16 since x E M is primitive. Let xl = x and write T = {x1}UT' = {xl}U{x'j}jEJ'. Let M1 = ({x'j}jEJ') and let N1 = M1f1N.
Claim. N = (slxl) ®N1. To see this, note that (six,) fl Nl C (x1) fl M1 = (0) because T is a basis of M. Let z E N. Then, with respect to the basis T, we may write z = alx1 +
b
jEJ'
Let d = (Si, al) = gcd{s1, al}. Then we may write d = us1 +val where u, V E R. If w = uy + vz, then Equation (6.5) shows that
w=uy+vz = (us1 + val)xl + E vbjx'j
jEJ'
= dx1 + = vbjxj. jEJ'
Writing w = c(w) w' where c(w) is the content of w and w' E M is primitive, it follows from Lemma 6.13 that c(w) I d (because c(w) is the greatest common divisor of all coefficients of w when expressed as a linear combination of any basis of M). Thus we have a chain of ideals (Si) C (d) C (c(w)),
and the maximality of (sl) in S shows that (Si) = (c(w)) = (d). In particular, (Si) = (d) so that s1 I a1, and we conclude that
z = b1(slxl) + E bjij jEJ'
3.6 Free Modules over a PID
155
That is, z E (s1x1) + N1. Theorem 3.1 then shows that
N = (six,) ® Ni, and the claim is proved. By Theorem 6.2, N1 is a free Rmodule since it is a submodule of the free Rmodule M. Furthermore, by the claim we see that
rank(N1) = rank(N)  1 = n  1. Applying the induction hypothesis to the pair N1 C M1, we conclude that there is a basis S' of M1 and a subset {x2, ... xn} of S', together with nonzero elements 82, ... , 8n of R, such that (6.6)
{82x2,
... , snxn }
is a basis of N1
and (6.7)
si I s;+1
for
2 < i < n  1.
Let S = S' U {x,}. Then the theorem is proved once we have shown tha Si 132.
To verify that s1 $2, consider the element 82x2 E Ni C N and let z = 81x1 + 32x2 E N. When we write z = c(z) z' where z' E M is primitive and c(z) E R is the content of z, Remark 6.15 shows that 1
c(z) = (s1, 82). Thus, (Si) C (c(z)) and the maximality of (s1) in S shows that (c(z)) = (s1), i.e., s1 I s2i and the proof of the theorem is complete. 0 (6.24) Example. Let N C Z2 be the submodule generated by y1 = (2, 4), y2 = (2, 2), and y3 = (2, 10). Then c(y1) = c(y2) = c(y3) = 2. Furthermore, 2 divides every component of any linear combination of y1, y2, and
y3, so s1 = 2 in the notation of Theorem 6.23. Let v1 = (1, 2). Then y1 = 2v1. Extend v1 to a basis of Z2 by taking v2 = (0,1). Then (6.8)
Ni = N n ((0, 1)) = ((0, 6)).
To see this note that every z E Ni can be written as z = a1y1 + a2y2 + a3y3
where a1, a3, a3 E Z satisfy the equation 2a1 + 2a2 + 2a3 = 0.
Thus, 4a1 = 4a2  4a3, and considering the second coordinate of z, we see that z = (z1, z2) where z2 = 4a1  2a2 + 10a3 = 6a2 4. 6a3 = 6(a3  a2).
Therefore, {v1i V2} is a basis of Z2, while {2v1, 6v2} is a basis of N. To check, note that y1 = 2v1, y2 = 2v1  6v2, and y3 = 2v1 + 6v2.
156
Chapter 3. Modules and Vector Spaces
(6.25) Remark. In Section 3.7, we will prove that the elements {s1, ... , sn}
are determined just by the rank n submodule N and not by the particular choice of a basis S of M. These elements are called the invariant factors of the submodule N in the free module M.
3.7 Finitely Generated Modules over PIDs The invariant factor theorem for submodules (Theorem 6.23) gives a complete description of a submodule N of a finitely generated free Rmodule M over a PID R. Specifically, it states that a basis of M can be chosen so
that the first n = rank(N) elements of the basis, multiplied by elements of R, provide a basis of N. Note that this result is a substantial generalization of the result from vector space theory, which states that any basis of a subspace of a vector space can be extended to a basis of the ambient space. We will now complete the analysis of finitely generated Rmodules (R a PID) by considering modules that need not be free. If the module M is not free, then, of course, it is not possible to find a basis, but we will still be able to express M as a finite direct sum of cyclic submodules; the cyclic submodules may, however, have nontrivial annihilator. The following result constitutes the fundamental structure theorem for finitely generated modules over principal ideal domains.
(7.1) Theorem. Let M # 0 be a finitely generated module over the PID R. If u(M) = n, then M is isomorphic to a direct sum of cyclic submodules
MRw1ED .®Rwn such that (7.1)
R # Ann(wl) J Ann(w2) D . . . D Ann(wn) = Ann(M).
Moreover, for 1 < i < n X7.2)
Ann(wi) = Ann (M/(Rwi}1 +
+ Rwn)) .
Proof. Since µ(M) = n, let {vi, ... vn} be a generating set of M and ,
define an Rmodule homomorphism 0: Rn + M by n O(a1,
... , an) _ E aivi. i=1
Let K = Ker(0). Since K is a submodule of Rn, it follows from Theorem 6.2 that K is a free Rmodule of rank in < n. By Theorem 6.23, there is a basis {yl, ... , yn} of Rn and nonzero elements sl, ... , s,n E R such that
3.7 Finitely Generated Modules over PIDs
{81y,,
(7.3)
157
is a basis for K
... , smym}
and si I si+l
(7.4)
for
1 < i < m  1.
Let w; = ¢(yi) E M for 1 < i < n. Then {w1, ... ,wn} generates M since 0 is surjective and {yl, ... , yn } is a basis of
Rn. We claim that
By the characterization of direct sum modules (Theorem 3.1), it is sufficient
to check that if aiwi +.... + anwn = 0
(7.5)
where ai E R, then aiwi = 0 for all i. Thus suppose that Equation (7.5) is satisfied. Then
ai (yl) + ... + anO(yn) = O(alyl + ... + anyn)
so that a1y1 + ... +anyn E Ker(O) = K = (sly,, ... , smym) Therefore,
aly, b1sly1 for some bl, ... , b,n E R. But {yl, ... , yn} is a basis of Rn, so we conclude that a; = bisi for 1 < i < m while ai =0 form + 1 < i < n. Thus, aiwi = bisiO(yi) = biO(siyi) = 0
for 1 < i < m because siyi E K = Ker(O), while aiwi = 0 for m+1 < i < n since ai = 0 in this case. Hence
MRw1®...®Rwn. Note that Ann(wi) _ (si) for 1 < i < m, and since si I si+,, it follows that
Ann(wl) 2 Ann(w2)
Ann(w,n),
while for i > m, since (yi) fl Ker(O) = (0), it follows that Ann(wi) = (0). Since si I sn for all i and since Ann(wi) = (si), we conclude that s,,M = 0. Hence, Ann(wn) = (sn) = Ann(M) and Equation (7.1) is satisfied. Since
Equation (7.2) follows from Equation (7.1). The proof is now completed
by observing that Ann(wi) 54 R for any i since, if Ann(wi) = R, then
158
Chapter 3. Modules and Vector Spaces
Rwi = (0), and hence, M could be generated by fewer than n elements. But n = µ(M), so this is impossible because µ(M) is the minimal number of generators of M. A natural question to ask is to what extent is the cyclic decomposition provided by Theorem 7.1 unique. Certainly, the factors themselves are not unique as one can see from the example Z2
Z.(1,0)ED More generally, if M is a free Rmodule of rank n, then any choice of basis {v1, ... ,v,a} provides a cyclic decomposition
Rv with Ann(vi) = 0 for all i. Therefore, there is no hope that the cyclic factors
themselves are uniquely determined. What does turn out to be unique, however, is the chain of annihilator ideals
Ann(wl) 2 ... where we require that Ann(w;) # R, which simply means that we do not allow copies of (0) in our direct sums of cyclic submodules. We reduce the uniqueness of the annihilator ideals to the case of finitely generated torsion Rmodules by means of the following result. If M is an Rmodule, recall that the torsion submodule M, of M is defined by M, = {x E M : Ann(x) 34 (0)}
(7.2) Proposition. If M and N are finitely generated modules over a PID R, then Ma, N if and only if M, N, and rank M/M, = rank N/N,.
Proof. Let 0 : M . N be an isomorphism. Then if x E M there is an a 94 0 E R with ax = 0. Then aO(x) = O(ax) = 0(0) = 0 so that O(x) E N,. Therefore, 0(M,) C N,. Applying the same observation to 0' chows that O(M,) = N,. Thus, OIM, : M, ' N, is an isomorphism; if n : N . N/N, is the natural projection, it follows that Ker(ao0) = M,. The first isomorphism theorem then gives an isomorphism M/M, = N/N,. Since M/M, and N/N, are free Rmodules of finite rank, they are isomorphic if and only if they have the same rank. The converse follows from Corollary 6.20.
Therefore, our analysis of finitely generated Rmodules over a PID R is reduced to studying finitely generated torsion modules M; the uniqueness of the cyclic submodule decomposition of finitely generated torsion modules is the following result.
3.7 Finitely Generated Modules over PIDs
159
(7.3) Theorem. Let M be a finitely generated torsion module over a PID R, and suppose that there are cyclic submodule decompositions
MRwi
(7.6)
and (7.7) where
(7.8)
Ann(wl) D
Ann(wk)
(0)
with
Ann(wl) 0 R
an d (7.9)
Ann(zl)
D Ann(zr.) # (0)
with
Ann(zi) # R.
Then k = r and Ann(wi) = Ann(zi) for 1 < i < k. Proof. Note that Ann(M) = Ann(wk) = Ann(zr). Indeed,
Ann(M) = Ann(Rwi + = Ann(wi) fl
+ Rwk) fl Ann(wk)
= Ann(wk)
since Ann(wi) D Ann(wk). The equality Ann(M) = Ann(zr) is the same argument. We will first show that k = r. Suppose without loss of generality that k > r. Choose a prime p E R such that (p) Ann(wi), i.e., p divides the generator of Ann(wi). Then (p) D Ann(wi) for all i. Since p E Ann(M/pM), it follows that M/pM is an R/pRmodule and Equations (7.6) and (7.7) imply (7.10)
MIpM
Rwi/(pRwi) ®...
Rwk/(pRwk)
and (7.11)
M/pM = Rzi/(pRzi) ® ... ® Rz,./(pRz,.).
Suppose that pRwi = Rwi. Then we can write apwi = wi for some a E R.
Hence, ap  1 E Ann(wi) C (p) by our choice of p, so 1 E (p), which contradicts the fact that p is a prime. Therefore, pRwi # Rwi for all i and Equation (7.10) expresses the R/pRmodule M/pM as a direct sum of cyclic R/pRmodules, none of which is (0). Since R/pR is a field (in a PID prime ideals are maximal), all R/pRmodules are free, so we conclude that M/pM is free of rank k. Moreover, Equation (7.11) expresses M/pM as a direct sum of r cyclic submodules, so it follows that k = µ(M/pM) < r.
Thus, r = k, and in particular, Rzi/(pRzi) 0 0 since, otherwise, M/pM could be generated by fewer than k elements. Thus, (p) 2 Ann(i) for all i; if not, then (p) + Ann(zi) = R, so there are a E R and c E Ann(zi) such that ap + c = 1. Then zi = apzi + czi = apzi E pRzi, so Rzi/(pRzi) = 0, and we just observed that Rzi/(pRzi) 54 0.
Chapter 3. Modules and Vector Spaces
160
We are now ready to complete the proof. We will work by induction
on t(Ann(M)) where, if I = (a) is an ideal of R, then t(I) is the number of elements (counted with multiplicity) in a prime factorization of a. This number is well defined by the fundamental theorem of arithmetic for PIDs. Suppose that t(Ann(M)) = 1. Then Ann(M) = (p) where p E R is prime. Since Ann(M) = Ann(wk) = Ann(zk) = (p) and since (p) is a maximal ideal, Equations (7.8) and (7.9) imply that Ann(wi) = (p) = Ann(zi) for all i, and the theorem is proved in the case t(Ann(M)) = 1. Now suppose the theorem is true for all finitely generated torsion Rmodules N with t(Ann(N)) < t(Ann(M)), and consider the isomorphisms (7.12)
pM'5 pRwl ® ... ® pRwk
pRw,+i ® ... ® pRwk
and (7.13)
pM L' pRzl ® ... ® pRzk =' pRzt+i ® ... ® pRzk
where Ann(wj) = ... = Ann(w,) = Ann(zi) = ... = Ann(zi) = (p) and Ann(w,+i) g6 (p), Ann(zt+i) 0 (p) (s and t may be 0). Then Ann(pM) = (a/p) where Ann(M) = (a), so t(Ann(pM)) = t(Ann(M))1. By induction we conclude that k  s = k  t, i.e., s = t, and Ann(pwi) = Ann(pzi) for s < i < k. But Ann(pwi) = (ai/p) where Ann(wi) = (ai). Thus Ann(wi) _ 0 Ann(zi) for all i and we are done.
R/ Ann(wi) and since R/I and R/J are isomorphic RSince Rwi modules if and only if I = J (Exercise 10), we may rephrase our results as follows.
(7.4) Corollary. Finitely generated modules over a PID R are in onetoone correspondence with finite nonincreasing chains of ideals
R#I,212_J...2In. Such a chain of ideals corresponds to the module
M=R/Il®...®R/In. Note that µ(M) = n and if Ik+i
In = (0) but Ik 96 (0), then
M^_'R/II®...®R/Ik®R"k We will use the convention that the empty sequence of ideals (n = 0) corresponds to M = (0). Proof
0
(7.5) Definition. If M is a finitely generated torsion module over a PID R and M Rwi ® ®Rwn with Ann(wi) Ann(wi+i) (1 < i < n  1) and Ann(wi) 0 R, then the chain of ideals Ii = Ann(wi) is called the chain of invariant ideals of M.
3.7 Finitely Generated Modules over PII)s
161
Using this language, we can express our results as follows:
(7.6) Corollary. Two finitely generated torsion modules over a PID are isomorphic if and only if they have the same chain of invariant ideals.
0
Proof.
(7.7) Remark. In some cases the principal ideals Ann(w3) have a preferred
generator aj. In this case the generators {ajIj=1 are called the invariant factors of M. The common examples are R = Z, in which case we choose aj > 0 so that aj = IZ/Ann(w,)I, and R = F[X], where we take monic polynomials as the preferred generators of ideals. (7.8) Definition. Let R be a PID, and let M be a finitely generated torsion Rmodule with chain of invariant ideals (31) ? (92) ? ... D (3n)
We define me(M) = sn and co(M) = 81
sn.
Note that me(M) and co(M) are only defined up to multiplication by a unit, but in some cases (R = Z or R = F[X]) we have a preferred choice of generators of ideals. In these cases me(M) and co(M) are uniquely defined. Concerning the invariants me(M) and co(M), there is the following trivial but useful corollary of our structure theorems.
(7.9) Corollary. Let M be a finitely generated torsion module over a PID R. (1) If a E R with aM = 0, then me(M) I a.
(2) me(M) divides co(M). (3) If p E R is a prime dividing co(M), then p divides me(M).
Proof. (1) Since Ann(M) = (sn) = (me(M)) by Theorem 7.1 and the defintion of me(M), it follows that if aM = 0, i.e., a E Ann(M), then me(M) I a. (2) Clearly s,, divides s1 ... sn. (3) Suppose that p I 31 sn = co(M). Then p divides some si, but (si) 2 (sn), so si I sn. Hence, p I sn = me(M). 0
(7.10) Remark. There are, unfortunately, no standard names for these invariants. The notation we have chosen reflects the common terminology in
the two cases R = Z and R = F[X]. In the case R = Z, me(M) is the exponent and co(M) is the order of the finitely generated torsion Zmodule
162
Chapter 3. Modules and Vector Spaces
(= finite abelian group) M. In the case R = FIX] of applications to linear algebra to be considered in Chapter 4, me(VT) will be the minimal polynomial and co(VT) will be the characteristic polynomial of the linear transformation T E HomF (V) where V is a finitedimensional vector space over the field F and VT is the FIX]module determined by T (see Example 1.5 (12)).
There is another decomposition of a torsion Rmodule M into a direct sum of cyclic submodules which takes advantage of the prime factorization of any generator of Ann(M). To describe this decomposition we need the following definition.
(7.11) Definition. Let M be a module over the PID R and let p E R be a prime. Define the pcomponent Mp of M by M p = {x E M : Ann(x) = (p")
for some
n E Z+}.
If M = Mp, then M is said to be pprimary, and M is primary if it is pprimary for some prime p E R. It is a simple exercise to check that submodules, quotient modules, and direct sums of pprimary modules are pprimary (Exercise 54).
(7.12) Theorem. If M is a finitely generated torsion module over a PID R, then M is a direct sum of primary submodules.
Proof. Since M is a direct sum of cyclic submodules by Theorem 7.1, it is sufficient to assume that M is cyclic. Thus suppose that M = (x) and suppose that Ann(x) = (a) = (pr' ... pr,,) where pl, ... , pn are the distinct prime divisors of a. Let qi = a/p;'. Then 1 = (ql, ... , qn) = gcd{ql, ... , q,,}, so there are bl, ... , bn E R such that (7.14)
1 = bigi + ... + bngn
Let xi = bigix. Then Equation (7.14) implies that
x=xl +...+ xn so that Al = (x) Suppose that y E (x1) fl ((x2) +
(XI) + ... + (xn) + (x,)). Then
y=Clx1
C2x2+...+Cnxn
and hence, p 'y = clb1Pi'gix = c1blax = 0 and qly = C2g2P23x2 +
+CngnPn"xn = 0,
3.7 Finitely Generated Modules over PIDs
163
Therefore, {pl' , ql } C Ann(y), but (pi' , ql) = 1 so that where q = ql Ann(y) = R. Therefore, y = 0. A similar calculation shows that
(xi) n ((xl) + ... + (xi) + ... + (xn)) = (0), so by Theorem
C
Combining Theorems 7.1 and 7.12, we obtain the following result:
(7.13) Theorem. Any finitely generated torsion module M over a PID R is a direct sum of primary cyclic submodules.
Proof. Suppose M Rwl® . ®Rwn as in Theorem 7.1. Then if Ann(wi) _ (si), we have si I si+1 for 1 < i < n  1 with sl 54 1 and sn # 0 (since M is torsion). Let p', ... pk be the set of distinct nonassociate primes that occur as a prime divisor of some invariant factor of M. Then Si =
ulpil,
... peklk
sn = unplnl ...
pknk
where the divisibility conditions imply that
0 < elf < e2j < ... < eni
for
1<j 0}, and the theorem is proved. submodules with annihilators 0
eij > 0, 1 < j < k} are called
(7.14) Definition. The prime powers the elementary divisors of M.
(7.15) Theorem. If M and N are finitely generated torsion modules over a
PID R, then M N if and only if M and N have the same elementary divisors.
Proof. Since M is uniquely determined up to isomorphism from the invariant factors, it is sufficient to show that the invariant factors of M can be recovered from a knowledge of the elementary divisors. Thus suppose that (sl)
(32)
...
(sn)
is the chain of invariant ideals of the finitely generated torsion module M. This means that si ( si+1 for 1 < i < n. Let pl, ... ,pk be the set of distinct nonassociate primes that occur as a prime divisor of some invariant factor of M. Then
164
Chapter 3. Modules and Vector Spaces
$1 = U1P11" ... Pkik (7.15) P`"k
sn = unP1"' ...
k
where the divisibility conditions imply that (7.16)
< enj
0 < e1j < e2j
0. That is to say, x E MT if and only if o(x) < oo. Moreover, Ann(x) = (n) means that o(x) = n. Thus the torsion submodule of M consists of the set of elements of finite order. Furthermore, M is finitely generated and torsion if and only if M is a finite abelian group. Indeed, if M = (x1, ... , xk) then any x E M can be written x = nix1 + + nkxk where 0 < ni < o(xi) < oo f o r 1 < i < k. Therefore, IMI < n 1 o(xi). Hence, the fundamental structure theorem for finitely generated abelian groups takes the following form.
(7.22) Theorem. Any finitely generated abelian group M is isomorphic to Zr ®M1 where IMi I < oo. The integer r is an invariant of M. Any finite abelian group is a direct sum of cyclic groups of prime power order and these prime power orders, counted with multiplicity, completely characterize
3.7 Finitely Generated Modules over PIDs
169
the finite abelian group up to isomorpism. Also any finite abelian group is uniquely isomorphic to a group Z,, X . . X Z,k where 8i 1 si+i for all i.
Proof.
Given a natural number n it is possible to give a complete list of all abelian groups of order n, up to isomorphism, by writing n = pi' Pkk where pl, ... ,pk are the distinct prime divisors of n. Let M be an abelian group of order n. Then we may write M as a direct sum of its primary components M^='MM ED ...ED Mpk
where IMp, I
Then each primary component Mp, can be written as a
direct sum
Mp,?'Zp',,® ®Zp:u where
1<ej,:5
e;t 0. Show that each fm is irreducible in fact, prime) as an element of the ring Al. Now consider' onsider the function f (n) = 1 for all it > 0. Show that f does not have a factorization into irreducible elements in Al. (Hint: It may help to think of f as the "infinite monomial" Xo ro)Xi (1)
36.
... Xm() ....
(Compare this exercise with Example 2.5.15.) Let F be a field, and let
2 = {p,(X) : pa(X) is an irreducible monic polynomial in FIXJ}.
We will say that a rational function h(X) = f (X)/g(X) E F(X) is proper if deg(f (X)) < deg(g(X)). Let F(X)pr denote the set of all proper rational functions in F[XJ. (a) Prove that F(X) (b) Prove that B=
FIX] tD F(X)pr as Fmodules.
X'
E Z; 0 < j < deg(pa(X)), k > 1 }
(p. (X))'
is a basis of F(X)pr as an Fmodule. The expansion of a proper rational
function with respect to the basis B is known as the partial fraction
37. 38.
expansion; it should be familiar front elementary calculus. Prove that Q is not a projective Zmodule. Let
R = { f : [0, 1]  R :
f is continuous and f (0) = f (1) }
and let M = {f : 10, 11
R:
f is continuous and f (0) =  f (1)}.
Then R is a ring under addition and multiplication of functions, and M is an Rmodule. Show that M is a projective Rmodule that is not free. (Hint: 39.
Show that M ®M ? , R (D R.) Show that submodules of projective modules need not be projective. (Hint: Consider pZP2 C Z,2 as ZD2modules.) Over a PID, show that submodules of projective modules are projective.
40.
(a) If R is a Dedekind domain, prove that R is Noetherian.
b If R is an integral domain that is a local ring (i.e., R has a unique
maximal ideal), show that any invertible ideal I of R is principal. (c) Let R be an integral domain and S C R \ {0} a multiplicatively closed subset. If I is an invertible ideal of R, show that Is is an invertible ideal of Rs.
3.9 Exercises
179
(d) Show that in a Dedekind domain R, every nonzero prime ideal is maximal. (Hint: Let M be a maximal ideal of R containing a prime ideal P, 41. 42. 43.
44.
and let S = R \ M. Apply parts (b) and (c).) Show that Z[f3] is not a Dedekind domain. Show that Z[XJ is not a Dedekind domain. More generally, let R be any integral domain that is not a field. Show that RIX] is not a Dedekind domain. Suppose R is a PID and M = R(x) is a cyclic Rmodule with Ann M = (a) 9 (0). Show that if N is a submodule of M, then N is cyclic with Ann N = (b) where b is a divisor of a. Conversely, show that M has a unique submodule N with annihilator (b) for each divisor b of a.
Let R be a PID, M an Rmodule, x E M with Ann(x) = (a) 96 (0). Factor
a = upi' ... pk" with u a unit and pi, ..., pi, distinct primes. Let Y E M with Ann(y) = (b) 0 (0), where b = ti p1`' . . .pk ' with 0 < rn; < n; for 1 < i < k. Show that Ann(x + y) = (a). Let R be a PID, let M be a free Rmodule of finite rank, and let N C M be a submodule. If M/N is a torsion Rmodule, prove that rank(M) = rank(N). 46. Let R be a PID and let M and N be free Rmodules of the same finite rank. Then an Rmodule homomorphism f : M  N is an injection if and only if N/ Im(f) is a torsion Rmodule. 45.
47.
Let u = (a, b) E Z2.
(a) Show that there is a basis of Z2 containing u if and only if a and b are relatively prime. (b) Suppose that u = (5,12). Find a v E Z2 such that {u, v} is a basis of
V.
48. Let M be a torsion module over a PID R and assume Ann(M) = (a) 0 (0).
If a = pi' pk'' where pi, ... , pk are the distinct prime factors of a, then show that MD, = qiM where q; = a/p;'. Recall that if p E R is a prime,
then M9 denotes the pprimary component of M. Let M be a torsionfree Rmodule over a PID R, and assume that x E M is a primitive element. If px = qx' show that q I p. 50. Find a basis and the invariant factors for the submodule of Z3 generated by 49.
x1 _ (1,0,1), x2 = (4,3, 1), x3 = (0,9,3), and x4 = (3, 12,3).
51.
Find a basis for the submodule of Q(XJ3 generated by
f, = (2X1,X,X2+3), 52.
f2 = (X,X,X2),
f3 = (X + 1, 2X, 2X2 3).
Determine the structure of Z3/K where K is generated by xl = (2,1, 3) and x2 = (1,1,2).
Let R = RIXJ and suppose that M is a direct sum of cyclic Rmodules with annihilators (X  1)3, (X2 + 1)2, (X  1)(X2 + 1)4, and (X + 2)(X2 + 1)2. Determine the elementary divisors and invariant factors of M. 54. Let R be a PID and let p E R be a prime. Show that submodules, quotient modules, and direct sums of pprimary modules are pprimary. 53.
55. An Rmodule M is said to be irreducible if (0) and M are the only submodules of M. Show that a torsion module M over a PID R is irreducible if and only if M = R(x) where Ann(x) = (p) where p is prime. Show that
if M is finitely generated, then M is indecomposable in the sense that M is not a direct sum of two nonzero submodules if and only if M = R(x) where Ann(x) = (0) or Ann(z) = (p°) where p is a prime. 56. Let M be an Rmodule where R is a PID. We say that M is divisible if for each nonzero a E R, aM = M. (a) Show that Q is a divisible Zmodule.
Chapter 3. Modules and Vector Spaces
180
57. 58. 59. 60.
(b) Show that any quotient of a divisible Rmodule is divisible. It follows for example that Q /Z is a divisible Zmodule. (c) If R is not a field, show that no finitely generated Rmodule is divisible. Determine all nonisomorphic abelian groups of order 360. Use elementary divisors to describe all abelian groups of order 144 and 168. Use invariant factors to describe all abelian groups of orders 144 and 168. If p and q are distinct primes, use invariant factors to describe all abelian groups of order
(a) p2g2, (b) p" q, (c) P5. 61. If p and q are distinct primes, use elementary divisors to describe all abelian groups of order p3q2.
62.
63. 64.
65. 66.
67.
Let G, H, and K be finitely generated abelian groups. If G x K °_° H x K, show that G I H. Show by example that this need not be true if we do not assume that the groups are finitely generated. Determine all integers for which there exists a unique abelian group of order n. Show that two finite abelian groups are isomorphic if and only if they have the same number of elements of each order. Let p be a prime and assume that a finite abelian group G has exactly k elements of order p. Find all possible values of k Find a generator for the cyclic group F' where F is each of the following fields (see Example 2.5.15 (3)): (a) F2(XJ/(X2 + X + 1). (b) F3[X]/(X2 + 1). Let 0
M1

M2
/s+
... " Mn+l + 0
be an exact sequence of finite rank free modules and homomorphisms over a
PID R. That is, fl is injective, fn is surjective, and Im(f1) = Ker(f;+l) for
1 1. One may also speak of nlinearity on columns, but Proposition 2.9 will show that there is no generality gained in considering both types of nlinearity. Therefore, we shall concentrate on rows. (2.2) Examples.
(1) Let Dl and D2 be nlinear functions. Then for any choice of a and b in R, the function D : Mn(R) R defined by D(A) = aDi(A)+bD2(A) is also an nlinear function. That is, the set of nlinear functions on Mn(R) is closed under addition and scalar multiplication of functions, i.e., it is an Rmodule.
(2) Let o E S. be a permutation and define D, : Mn(R)
R by the
formula
D0(A) = al(l)a2(2) ... ano(n)
(3)
where A = [a,3). It is easy to check that D, is an nlinear function, but it is not a determinant function since it is not alternating. Let f : Sn # R be any function and define Df : Mn(R) + R by the formula D f = EaES. f (a)D,. Applying this to a specific A = [a13] E Mn(R) gives
Df(A) =
aES
f (a)al o(,)a2a(2) .
ano(n).
By examples (1) and (2), Df is an nlinear function. (4) If n = 2 and c E R, then Dc(A) = c(a11a22  a12a21) defines a deter
minant function on M2(A). The first order of business is to prove that there is a determinant function for every n and for every commutative ring R and that this determinant function is essentially unique. More precisely, any determinant function is completely determined by its value on the identity matrix In. Note that for n = 1 this is clear since D([a]) = D(a[1]) = aD([1]) for every 1 x 1 matrix [a] by property (1) of nlinearity. The strategy for verifying existence and essential uniqueness for determinant functions is to first verify a number
of basic properties that any determinant function must satisfy and then from these properties to derive a formula that must be used to define any determinant function. It will then only remain to check that this formula, in fact, defines a determinant function.
196
Chapter 4. Linear Algebra
(2.3) Lemma. Let D : M,,(R)  R be nlinear. If rowi(A) = 0 for some i, then D(A) = 0. Proof. Since rowi(A) = 0 rowi(A), property (1) of nlinearity applies to show that D(A) = 0 D(A) = 0. 17 (2.4) Lemma. Let D : M,, (R) a R be a determinant function. If i 96 j and Pij is the elementary permutation matrix determined by i and j, then
D(PijA) = D(A) for all A E
(That is, interchanging two rows of a matrix multiplies
D(A) by 1.) Proof. Let Ak = rowk(A) for 1 < k < n, and let B be the matrix with rowk(B) = rowk(A) whenever k 0 i, j while rowi(B) = rowj(B) = Ai+Aj. Then since D is nlinear and alternating,
Ai+Aj
Aj
=D
0=D(B)=D
+D Ai+Aj
Aj + Aj
=D
+D
+D Aj
Ai
=D
Aj
Ai
Ai
Aj
+D
= D(A) + D(PijA) by Proposition 1.12. Thus, D(PijA) = D(A), and the lemma is proved. 0
(2.5) Remark. Lemma 2.4 is the reason for giving the name "alternating"
to the property that D(A) = 0 for a matrix A that has two equal rows. Indeed, suppose D has the property given by Lemma 2.4, and let A be a matrix with rows i and j equal. Then PijA = A, so from the property of
4.2 Determinants and Linear Equations
197
Lemma 2.4 we conclude D(A) = D(A), i.e., 2D(A) = 0. Thus, if R is a ring in which 2 is not a zero divisor, the two properties are equivalent, but in general the property of being alternating (as given in Definition 2.1) is stronger (Exercise 16).
(2.6) Lemma. Let D : Mn(R)  R be a determinant function. If i 34 j, a E R, and Tii (a) is an elementary transvection, then
D(Tii(a)A) = D(A). (That is, adding a multiple of one row of A to another row does not change the value of D(A).)
Proof. Let B be the matrix that agrees with A in all rows except row i, and assume that row; (B) = or rowi (A). Let A' be the matrix that agrees with A in all rows except row i and assume that rowi(A') = rowi(A). Then D is alternating so D(A') = 0 since rowi(A') = rowi(A) = rowi(A'), and thus, D(B) = aD(A') = 0 since D is nlinear. Since Tij(a)A agrees with A except in row i and rowi(Tii(a)A) = rowi(A) + a rowi(A) = rowi(A) + rowi(B) (see Proposition 1.12), it follows from property (2) of nlinearity that
D(Tii (a)A) = D(A) + D(B) = D(A). 0
Let E1 = row; (In) for 1 < i < n, and consider all n x n matrices formed by using the matrices E; as rows. To develop a convenient notation, let Stn denote the set of all functions w : {1, 2, ..., n} {1, 2, ..., n},
and if w E Stn, let P,,, denote the n x n matrix with rowi(P,) = E,(;). For example, if n = 3 and w(1) = 2, w(2) = 1, and w(3) = 2, then P,,, =
0
1
0
1
0
0
0
1
0
If W E fln is bijective so that w E Sn, then P,, is called a permutation matrix. If w = (i j) is a transposition, then P,, = Pii is an elementary permutation matrix as defined in Section 4.1. In general, observe that the product ,,,A is the matrix defined by rowi(P,A) = row,,(,)(A).
According to Proposition 1.12 (3), if w E Sn is written as a product of transpositions, say w = (i1 jl)(i2 j2) ... (it it), then PW
= Pilii Pisi2 ... Pi9i, In = InPiiit Pi2iz ... Piti,
198
Chapter 4. Linear Algebra
Since w1 = (it jt)(iti jti)
(ii j1), the second equality, together with
Proposition 1.13 (3), shows that
coli(P.) = E,.,,(i) Therefore, again using Proposition 1.13 (3), we see that AP,, is the matrix defined by coli(AP,,) = col,,,(i)(A). To summarize, left multiplication by P,,, permutes the rows of A, following the permutation w, while right multiplication by P,,, permutes the columns of A, following the permutation w1.
Recall that the sign of a permutation a, denoted sgn(o), is +1 if a is a product of an even number of transpositions and 1 if a is a product of an odd number of transpositions. (2.7) Lemma. Let R be a commutative ring and let D : MM(R) . R be a determinant function. If w E Stn, then (1) D(P,,,) = 0 if w V Sn, and (2)
D(P,,,) = sgn(w)D(In) if w E Sn.
Proof. (1) If w V Sn then w(i) = w(j) for some i $ j so that P, has two rows that are equal. Thus, D(P,,) = 0. (2) If w E Sn, write w = (i1 ii) to get (by Proposition 1.12)
(it it) as a product of transpositions
Pw=Pi,i,...piti, By Lemma 2.4, we conclude that
(1)tD(In) = sgn(w)D(In), 0
and the lemma is proved.
Now let A = [aii] E Aln(R) and partition A by its rows, i.e., Al A
An
where Ai = rowi(A) for 1 < i < n. Note that Ai = [ail ... ain[
= [ail 0 ... 0) + 10 ail 0 ... 01+ ... + [0 ... pain] = ail E1 + ai2E2 + ... + ainEn n
_ >aijEj, j=1
4.2 Determinants and Linear Equations
199
and thus, Fn En,=1 aij1Ej1 E uj2=1 a2j2Ej3
A =
`n `', 1 anjn Ejn If D is a determinant function, we may compute D(A) using (2.1), the nlinearity of D, and Lemma 2.7 as follows:
En En =1 a1j1 Ej1 E a2j3Ejs D(A) = D uj2 [n
Lj
anjn Ejn
Ej2=1Ej1
n
a2j2Ejs
alj, D j1=1
ujn n
anjnEjn
Ej,
n j2
a1j1a2j3D 11=1 j,=1
En
nc 1 anjnEjn
n
n
= F ... E a1j1 a2jz ... anjn D ji=1
Ej1 Ej2
jn=1 Ejn
1: a1 (1)a2,a(2)...anw(n)D(PW) LaE12n
(2.2)
=
L:
sgn (w)a1,,(1) ...
l an,,()] D(In)
Thus, we have arrived at the uniqueness part of the following result since formula (2.2) is the formula which must be used to define any determinant function.
(2.8) Theorem. Let R be a commutative ring and let a E R. Then there is exactly one determinant function D. : Mn(R)  R such that Da(In) = a. Thus D. = aD1, and we let det = D1.
Proof. If there is a determinant function D then according to Equation (2.2), Da must be given by
Chapter 4. Linear Algebra
200
(2.3)
D0(A) = a E sgn(o )a1 a(1) ...ano(n)
I
J
oES
It remains to check that Equation (2.3) in fact defines a determinant func
tion on Mn(R). It is sufficient to check this with a = 1 since a scalar multiple of a determinant function is also a determinant function. If we let
f (a) = sgn(a) 1, then Dl = Df as defined in Example 2.2 (3), and as observed in that example, Df is an nlinear function for each f : Sn + R. In particular, Dl is nlinear and it remains to check that it is alternating. To verify this, suppose that A E Mn(R) has row, (A) = rowj(A) with i 96 j.
Then for 1 < k < n we have aik = ajk. If a E Sn, let a' = a o (i j). We claim that (2.4)
al (1) "' ano(n) = a, ,(1) "' ano'(n)
This is because a(k) = a'(k) if k 54 i, j so that ako(k) = ako'(k), while a'(i) = a(j) and a'(j) = a(i) so that a,,,(,) = a,,(j) = aj,(j) and aj,'(j) = aj(,) = a,,(;). Hence Equation (2.4) is valid. But sgn(a') =  sgn(a), so sgn(a)al ,(,) ... ano(n) + sgn(a')al,,(,) ... an al(n) = 0.
But a
a' gives a pairing of the even and odd permutations in Sn, and
hence, we conclude
sgn(a)al 0(1) ... an v(n)
D1(A) = oES
=
(sgn(a)al0(1)... ano(n) + sgn(a')al,,(1) ... ano'(n))
egn(o)=1
=0. Therefore, Dl is a determinant function on Mn(R) and Theorem 2.8 is proved.
(2.9) Corollary. Let A = [aij] E Mn(R) be an upper (resp., lower) triangular
matrix, i.e., a,j = 0 for i > j (reap., i < j). Then n
det(A) = all ... ann = flail. i=1
Proof. If a E Sn is not the identity permutation, then for some i and j, a(i) > i and a(j) <j, so in either case, a,,(1)...an0(n) = 0,
and the result follows by Equation (2.3).
4.2 Determinants and Linear Equations
201
Our determinant functions have been biased towards functions on the rows of matrices. But, in fact, we can equally well consider functions on the columns of matrices. Perhaps the simplest way to see this is via the following result:
(2.10) Proposition. If A E Mn(R) then det(A) = det(At). Proof. Let A = [aid). Then
det(A) _ > sgn(a)a1 ,(1) ... ano(n) oES., sgn(a)ao1(,(1))c(1)...ao1(C(R))C\f1)
oES,
_
1:
sgn(a)ao_,(1) 1 ...ao1(n)n
oES.,
E Sgn(a1)ao1(1) I ... so1(n) n cES sgn(r)a7(1) 1
... a1.(n) n
*ES.,
= det(Al).
Here we have used the fact that sgn(a1) = sgn(a) and that av1(o(1)) 0(1) "' ao1(o(n)) o(n) = ao,(1) 1 ... ao_,(n) n,
This last equation is valid because R is commutative and {a(1), ..., a(n)} 0 is just a reordering of {1, ..., n} for any a E Sn. (2.11) Theorem. If A, B E Mn(R) then
det(AB) = det(A) det(B). Proof. Define DB : M,, (R) . R by DB(A) = det(AB). Since rowi(AB) = [rowi(A)]B, it is easy to check (do it) that DB is nlinear and alternating. Thus DB is a determinant function, so by Theorem 2.8, DB(A) = adet(A) where a = DB(In) = det(B). 0
This result is an example of the payoff from the abstract approach to determinants. To prove Theorem 2.11 directly from the definition of determinant as given by Equation (2.3) is a somewhat messy calculation. However, the direct calculation from Equation (2.3) is still beneficial in that a more general product formula is valid. This approach will be pursued in Theorem 2.34.
(2.12) Corollary. If R is a commutative ring, A E Mn(R), and S E GL(n, R), then
202
Chapter 4. Linear Algebra
det(S'AS) = det(A). That is, similar matrices have the same determinant. Proof. Exercise.
0
Similarly to the proof of Theorem 2.11, one can obtain the formula for the determinant of a direct sum of two matrices.
(2.13) Theorem. If R is a commutative ring, A; E Mn; (R) for 1 < i < r, and A = ® 1A;, then r
det(A) = rl det(Ai). i=1
Proof. It is clearly sufficient to prove the result for r = 2. Thus suppose that A = A1® A2. We may write A = (A1®I,,,) (In, ® A2). Then Theorem 2.11 gives
det(A) = (det(A1 ® I,,,)) (det(In, ® A2))
.
Therefore, if det(A1 ® In,) = det(A1) and det(In, ® A2) = det(A2), then we are finished. We shall show the first equality; the second is identical. Define D1 : Mn1(R) + R by D1(B) = det(B (D In3). Since det is (n1 + n2)linear and alternating on Mn1+n2(R), it follows that D1 is n1linear and alternating on M,,,(R). Hence, (by Theorem 2.8), D1(A1) = adet(A1) for all Al E Mn1(R), where a = D1(In,) = det(In, (D In3) = 1, and the theorem is proved. 0
Remark. Theorem 2.13 and Corollary 2.9 can both be generalized to a formula for block triangular matrices. (See Exercise 13.) There is also a simple formula for the determinant of the tensor product of two square matrices.
(2.14) Proposition. Let R be a commutative ring, let A E Mm(R), and let B E Mn(R). Then det(A (9 B) = (det(A))n (det(B))m.
Proof. By Corollary 1.19, A ® B = (A (9 In)(I, ® B). By Example 1.17 (2), Im 0 B = ®i'_' 1B, so det(I,n ® B) = det(B)m by Theorem 2.13. We leave it as an exercise to verify that the rows and columns of the matrix A ® In can be permuted to obtain that A ® In is (permutation) similar to ® 1A (Exercise 50). The proof is then completed by another application of Theorem 2.13.
0
4.2 Determinants and Linear Equations
203
We will now consider cofactor expansions of determinants. If A E A,(R) and 1 < i, j < n, let Aid be the (n  1) x (n  1) matrix obtained by deleting the i1h row and jth column of A. (2.15) Theorem. (Laplace expansion) Let R be a commutative ring and let A E Mn(R). Then n
(1)
1:(1)k+'aki det(Ak,) = bid det(A) k=1
and n
(2)
>(_1)k+iaik det(A,k) = bij det(A). k=1
Proof. If A E Mn(R) and 1 < i, j < n, let n
Dii(A) = 1: (1)k+'akidet(Aki) k=1
That is, Di,(A) E R is defined by the lefthand side of equation (1). We claim that the function Did : Mn(R) . R is a determinant function for all i, j. Note that the function A '. akidet(Ak,) is nlinear on Mn(R). Since a linear combination of nlinear functions is nlinear, it follows that Did is nlinear. It remains to check that it is alternating. To see this, suppose that A E Mn(R) is a matrix with rows p and q equal. If k 96 p, q, then Ak; has two rows equal, so det(Ak,) = 0 in this case. Thus, (2.5)
Di,(A) = (1)p+"a, det(Apj) + (1)q+3agidet(Agi)
Note that the assumption that rowp(A) = rowq(A) means that api = aqi. To be explicit in our calculation, suppose that p < q. Then the matrix Aqj is obtained from App by moving row q  1 of Aye to row p and row t of App
to row t + l for t = p, ... , q2. In other words, Aqi
= PAp;
where w E Sn_1 is defined by w(t) = t fort < p and t > q, while w(q1) = p
and w(t) = t + 1 for t = p, ... , q  2. That is, w is the (q  p)cycle (p,p+1,...,q1).Then byLemma 2.7 det(Aq,) = det(PA) = sgn(w) det(Apj)
(1)9_p1
det(Ap,).
Equation (2.5) then shows that Dij (A) = 0, and hence Dij is alternating. Therefore, Did is a determinant function, so by Theorem 2.8 (2.6)
Dii(A) = Dii(I.)det(A)
204
Chapter 4. Linear Algebra
for all A E M,a(R). A direct calculation shows that Di1(In) = bid, so that Equation (2.6) yields (1) of the theorem. Formula (2) (cofactor expansion along row j) is obtained by applying formula (1) to the matrix At and using the fact (Proposition 2.10) that det(A) = det(A'). 0 (2.16) Definition. If A E MM(R), then we define the cofactor matrix of A, denoted Cofac(A), by the formula
enti,(Cofac(A)) _
(1)i+.i det(Aij),
and we define the adjoint of A, denoted Adj(A), by Adj(A) = (Cofac(A))t.
The following result should be familiar for matrices with entries in a field, and the proof is the same: (2.17) Theorem. Let A E Mn(R).
(1) A(Adj(A)) = (Adj(A))A = det(A)In. (2) A is invertible if and only if det(A) is a unit in R, and in this case,
A' = (det(A))' Adj(A). Proof. Formula (1) follows immediately from Theorem 2.15. If A E Mn(R)
is invertible, then there is a matrix B such that AB = BA = In. Thus (det(A))(det(B)) = 1 so that det(A) E R'. Conversely, if det(A) is a unit, then B = (det(A))' Adj(A) satisfies AB = BA = In by formula (1). 0 (2.18) Remark. The definition of inverse of a matrix requires that AB = BA = I,,, but as a consequence of Theorem 2.17, we can conclude that a matrix A E Mn(R) is invertible provided that there is a matrix B E MM(R)
such that AB = In. Indeed, if AB = In then (det(A))(det(B)) = 1, so det(A) is a unit in R, and hence, A is invertible. (2.19) Examples.
(1) A matrix A E Mn(Z) is invertible if and only if det(A) = ±1.
If F is a field, a matrix A E Mn(F[X]) is invertible if and only if det(A) E F' = F \ {0}. (3) If F is a field and A E Mn(F[X]), then for each a E F, evaluation of each entry of A at a E F gives a matrix A(a) E M,,(F). If det(A) = f (X) 34 0 E FIX], then A(a) is invertible whenever f (a) 0 0. Thus, (2)
A(a) is invertible for all but finitely many a E F.
4.2 Determinants and Linear Equations
205
As an application of our results on determinants, we shall present a determinantal criterion for the solvability of homogeneous linear equations
when the coefficients can be from an arbitrary commutative ring. This criterion will involve some ideals of R generated by various determinants of submatrices of matrices over R. We will start by defining these ideals.
To state the results in a reasonably compact form it is convenient to introduce some appropriate notation. If p, m E N with p _< m, let Qp,,,m denote the set of all sequences a = (i1 i ..., ip) of p integers with < ip < m. Note that the cardinality of the set Qp,m 1 < i1 < i2 < is IQp,ml = (p). Suppose A E M,,,,,,(R). If a E Qp,m and ,0 E Qj,n, let A[a I 0) denote the submatrix of A consisting of the elements whose row
index is in a and whose column index is in Q. If a E Qp,m then there is a complementary sequence a E Qp_m,m consisting of the integers in {1, 2,. .. , m} not included in a and listed in increasing order. To give some examples of these notations, let A =
all
a12
a13
a14
all
a22
a23
a24
a31
a32
a33
a34
If a = (1, 3) E Q2,3 and 0 = (2, 3) E Q2,4, then A[a 18]
a12 a32
a13
a33]
while a = 2 E Q1,3 and (3 = (1, 4) E Q2,4 so that A[& I)3j = [ a21
a24 } .
(2.20) Definition. If R is a commutative ring and A E Mm,n(R), then a t x t minor of A is the determinant of any submatrix A[a 1,0] where a E Qt,m, Q E Qt,,. The determinantal rank of A, denoted Drank(A), is the largest t such that there is a nonzero t x t minor of A.
(2.21) Definition. If R is any commutative ring, A E Mm,n(R), and 1 < t < min{m, n}, let Ft (A) = ({det A[a I
/31
: a E Qt,m, Q E Qt,n}) C R.
That is, Ft (A) is the ideal of R generated by all the t x t minors of A. We set Fo(A) = R and Ft (A) = 0 if t > min {m, n}. Ft (A) is called the tth_
Fitting ideal of A. The Laplace expansion of determinants (Theorem 2.15) shows that Ft+l(A) C Ft (A). Thus there is a decreasing chain of ideals
R=Fo(A)2F1(A)_JF2(A)D .... If R is a PID, then Ft(A) is a principal ideal, say Ft (A) = (dt(A)) where dt(A) is the greatest common divisor of all the t x t minors of A. In this case, a generator of Ft (A) is called the tthdeterminantal divisor of A.
206
Chapter 4. Linear Algebra
(2.22) Definition. If A E M,n,n(R), then the Mrank(A) is defined to be the largest t such that Ann(Ft(A)) = 0. (2.23) Remarks.
(1) Mrank(A) = 0 means that Ann(F1(A)) 96 0. That is, there is a nonzero a E R with aa,j = 0 for all aid. Note that this is stronger than saying that every element of A is a zero divisor. For example, if A = [ 2 3 ] E M1,2(Z6), then every element of A is a zero divisor in Z6 but there is
no single nonzero element of Z6 that annihilates both entries in the matrix. (2) If A E then Mrank(A) = n means that det(A) is not a zero divisor of R.
(3) To say that Mrank(A) = t means that there is an a # 0 E R with a D = 0 for all (t + 1) x (t + 1) minors D of A, but there is no nonzero
b E R which annihilates all t x t minors of A by multiplication. In particular, if det Ala 101 is not a zero divisor of R for some a E Q,,,n, (3 E Q,,,,, then Mrank(A) > s. (2.24) Lemma. If A E Mm,n(R), then
0 < Mrank(A) < Drank(A) < min{m, n}.
0
Proof. Exercise.
We can now give a criterion for solvability of the homogeneous linear equation AX = 0, where A E M,,,,n(R). This equation always has the trivial solution X = 0, so we are looking for solutions X # 0 E Mn,1(R).
(2.25) Theorem. Let R be a commutative ring and let A E Mm,n(R). The matrix equation AX = 0 has a nontrivial solution X 0 0 E Mn,1(R) if and only if Mrank(A) < n.
0, so choose b 0 0 E R with b Ft+1(A) = 0. Without loss of generality, we may assume that t < m since, if necessary, we may replace the system AX = 0 by an equivalent one (i.e., one with the same solutions) by adding some rows of zeroes to the bottom of A. If t = 0 then ba,, = 0 for all aid and we
Proof. Suppose that Mrank(A) = t < n. Then Ann(Ft+1(A))
b
. Then X A 0 E M,1(R) and AX = 0.
may take X = b
Thus, suppose t > 0. Then b Ann(Ft(A)) = 0, so bdet Ala 1010 0 for some a E Qt.+,,,13 E Qt,,, By permuting rows and columns, which does not affect whether AX = 0 has a nontrivial solution, we can assume a =
4.2 Determinants and Linear Equations
207
F o r 1 < i < t + 1 let E Qt,t+l where i indicates that i is deleted. Let di = (1)t+1+' det A[a 1,8j). Thus dl, ... dt+l are the cofactors of the matrix
Al = A[(1,...,t + 1) [ (1, ... ,t +
1)]
obtained by deleting row t + 1 and column i. Hence the Laplace expansion theorem (Theorem 2.15) gives 1t+i j=l aijdj = 0
1 < i < t,
(2.7)
(1,...,t,t+1)J t m, which is possible by our hypothesis ISI > µ(M) = m. Since
M=(w1,...,wm),wemaywrite vj = al jwl + ... + amjwm
with aij E R. Let A = faij] E Mm,n(R). Since n > m, it follows that Mrank(A) < m < n, so Theorem 2.25 shows that there is an X 36 0 E Mn,1(R) such that AX = 0. Then (2.8)
Exivo = Exo j=1
j=1
m
_ i=1
(atiwi) i=1
r n
(a*xi)
Wi
j=1=1
=0 since AX = 0. Therefore, S is Rlinearly dependent.
0
210
Chapter 4. Linear Algebra
(2.32) Corollary. Let R be a commutative ring, let M be an Rmodule, and let N C M be a free submodule. Then rank(N) < rank(M). Proof. If rank(M) = oo, there is nothing to prove, so assume that rank(M) _ m < oo. If rank(N) > m, then there is a linearly independent subset of M, namely, a basis of N, with more than m elements, which contradicts Propo
0
sition 2.31.
(2.33) Remark. Corollary 2.32 should be compared with Theorem 3.6.2, concerning submodules of free modules over a PID. Also, note that the condition that N he free is necessary (see Remark 3.6.5). The following result, known as the CauchyBinet theorem, generalizes the formula for the determinant of a product (Theorem 2.11) to allow for products of possibly nonsquare matrices. One use of this formula is to investigate the behavior of the rank of matrices under products. (2.34) Theorem. (CauchyBinet formula) Let R be a commutative ring and
let A E Mm,n(R), B E Mn,p(R). Assume that 1 < t < min{m, n, p} and let a E Qt,m, Q E Qt,p. Then
det(AB[a) /3)) _ E det(A[a [ d)det(B[7 [ p1). 'YEQt.n
Proof. Suppose that a = (a1, ..., at), 13 = (Qt, ..., 3t) and let C = AB[a 10). Thus, entij (C) = row., (A) colp, (B) n
at:, kbko, k=1
so that Ek=1 a.,kbk3,
... >k.1 aQIkbkI,
C=
.
Ek=1
a.,k40,
aa,kbkO,
Using nlinearity of the determinant function, we conclude that n
(2.9)
detC= E k,=1
bk, Q,
n
Ea.,k, ke_1
bk, Ot
a., k, det bkt A,
...
bkt At
If k; = kj for i 34 j, then the ith and jth rows of the matrix on the right are equal so the determinant is 0. Thus the only possible nonzero determinants on the right occur if the sequence (k1, ... , kt) is a permutation of a sequence
4.2 Determinants and Linear Equations
211
It) E Qt,,,. Let or E St be the permutation of {1, ..., t} such y = (yi, that y, = k,(,) for 1 < i < t. Then bkl 01
bkl 01
det
(2.10)
= sgn(a) det B[y 1,31. bk, a,
bk, Q,
Given y E Qt,,,, all possible permutations of y are included in the summation in Equation (2.9). Therefore, Equation (2.9) may be rewritten, using Equation (2.10), as
)detBIV
detC = 'YEQe ,..
I p]
eES
J
E det A[a I y] det B[y 10], 7EQ*,+.
which is the desired formula.
11
(2.35) Examples.
(1) The CauchyBinet formula gives another verification of the fact that det(AB) = det A det B for square matrices A and B. In fact, the only element of Q,,,,, is the sequence y = (1, 2, ... ,n) and A[y I y] = A, B[y yJ = B, and AB[y I y] = AB, so the product formula for determinants follows immediately from the CauchyBinet formula. (2) As a consequence of the CauchyBinet theorem, if A E Mm,n(R), and B E Mn,p(R) then I
(2.11)
Drank(AB) < min{Drank(A), Drank(B)}.
To see this, let t > min{Drank(A), Drank(B)} and suppose that a E Qt,,n, Q E Qt,p. Then by the CauchyBinet formula
det(AB[a 1,31) = E det(A[a I y]) det(B[y 10]). fEQt...
Since t > min{Drank(A), Drank(B)}, at least one of the determinants det A[a I y] or det B[y I a] must be 0 for each y E Qt,,,. Thus, det(AB[a 10]) = 0, and since a and 3 are arbitrary, it follows that Drank(AB) < t, as required. Equation (2.11) easily gives the following result, which shows that determinantal rank is not changed by multiplication by nonsingular matrices.
(2.36) Proposition. Let A E M..n(R) and let U E GL(m, R), V E GL(n, R). Then
212
Chapter 4. Linear Algebra
Drank(UAV) = Drank(A).
Proof Any matrix B E Mm,n(R) satisfies Drank(B) < min{m, n} and since Drank(U) = n and Drank(V) = m, it follows from Equation (2.11) that Drank(UAV) < min{Drank(A), n, m} = Drank(A) and
Drank(A) = Drank(U1(UAV)V1) < Drank(UAV).
0
This proves the result.
We will conclude this section by giving a version of the Laplace expansion theorem that allows for expansion of det A along a given set of rows rather than a single row. The choice of rows along which expansion takes place is given by an element a E Qt,,. Recall that if y E Qt,n then 7 E Q,,t,,, denotes the complementary sequence. With these preliminaries out of the way, the general Laplace expansion theorem can be stated as follows:
(2.37) Theorem. (Laplace expansion) Let A E Mn(R) and let a E Qt,n (1 t < n) be given. Then (2.12)
det A =
(1)e(o)+a(ti> det(A[a I y]) det(A(a
where
ry])
t
s(Y) = E yj j=1
for any y = (Y1, ... ,'Yt) E
Proof. The proof is essentially similar to the proof of Theorem 2.15 (which is a special case of the current theorem). If A E M,,(R), define (2.13)
D,,(A) = E (_i)88 (,) det(A(a I y])det(A[a 17]) 7EQt..
Then D,, : Mn(R)  R is easily shown to be nlinear as a function on the columns of A E Mn(R). To complete the proof, it is only necessary to show that D,, is alternating and that D ,(In) = 1. Thus, suppose that colp(A) = colq(A), and to be specific, assume that p < q. If p and q are both in y E Qt,n, then A(a I yJ will have two columns equal so that det A [a I y] = 0,
while, if both p and q are in 3 E Qn_t,n, then det A[& 17J = 0. Thus, in the evaluation of D,,(A) it is only necessary to consider those y E Qt,n such that p E y and q E ^y, or viceversa. Thus, suppose that p E y, q E ry and
4.2 Determinants and Linear Equations
213
define a new sequence 7' E Qt,,, by replacing p E 7 by q. Then ry' agrees with ry except that q has been replaced by p. Thus (2.14)
s(7')  s(7) = q  p
Now consider the sum
(1)°(7) det(A[a 17]) det(A[a 17]) + (1)"('Y') det(A[a 17']) det(A[a
which we denote by S(A). We claim that this sum is 0. Assuming this, since y and 7' appear in pairs in Qt,,,, it follows that DQ(A) = 0 whenever two columns of A agree; thus DQ is alternating. It remains to check that S(A) = 0.
Suppose that p = 7k and q = it. Then 7 and 7' agree except in the range from p to q, as dory and ry'. This includes a total of q  p + 1 entries. If r of these entries are included in 7, then 71 < ... < 7k = p < 7k+1 < ... < 7k+,1 < q < 7k+r < ... < 7t
and
A[aI7')=A[aI7]P,., where w is the rcycle (k + r  1, k + r  2, ... , k). Similarly,
A[&I7J=A[aI7JPW' where w' is a (q  p + 1  r)cycle. Thus, det(A[a 17']) det(A[a 17 J)
_
(1)°(1')+(r1)+(qD)r
det(A[a 171) det(A[a I
Yl)
Since s(7') + (q  p)  1  s(7) = 2(q  p)  1 is odd, we conclude that S(A) = 0. Thus DQ is alternating, and since it is straightforward to check that 1, the result is proved. 0 Applying formula (2.12) to At in place of A, gives the Laplace expansion in columns:
(2.15)
det A
=E
det(A[7 I aJ) det(A[ry I &]).
7EQt.
Note that if t = 1 then Ql,,, = {1, 2, ..., n} so that A[i I j] = a+i while Al? I j = A13, so Theorem 2.37 includes Theorem 2.15 as a special case.
214
Chapter 4. Linear Algebra
4.3 Matrix Representation of Homomorphisms Before beginning with the procedure of associating a matrix with a homomorphism between free modules, we would like to make some remarks about the ticklish situation that arises for a noncommutative ring. We will only need this once, at the very end of Section 7.1, so the reader who is only interested in the commutative case may (and is well advised to) skip the more general situation. Difficulties already arise in the simplest case. Let R be a ring and let us consider a free Rmodule M of rank I with basis 5 = {v}. Then M = {rv : r E R}. We wish to give "coordinates" to the elements of M, i.e., identify the elements of M with the elements of R, and clearly, there is only one reasonable choice here, namely, that, in the basis 8, rv should have coordinate 1rv[8 = [r[. Now consider f E EndR(M). We wish to represent
f by a matrix with respect to the basis 5, and again there is only one reasonable choice: if f (v) = sv, then f should have coordinate matrix in the basis 5 given by [f]13 = [s[. (Note that f is not "leftmultiplication by s" unless s E C(R), the center of R. Indeed, g : M M defined by g(m) = sm is not an Rendomorphism unless s E C(R), as then g(rm) = srm # rsmn = rg(m) in general. Of course, there is no problem if R is commutative.) Now, the theory we are about to develop will tell us that for any m E M, we may calculate f (m) by [f(m)1S = If1B[rn]a.
However, when we try to apply this to ni = rv, we get f (m) = f (rv) _ rf(v) = r(sv) = (rs)v, so [f(m)15 = [rs) while [f]a[m]a = [s][r] = [sr]. If R is commutative, these are equal, but in general they are not. On the other hand, this formulation of the problem points the way to its solution. Namely, recall that we have the ring R°P (Remark 3.1.2 (3)) whose elements are the elements of R, whose addition is the same as that of R, but whose multiplication is given by r s = sr, where on the righthand side we have the multiplication of R. Then, indeed, the equation [rs1 = Is] [r]
is valid, and we may hope that this modification solves our problem. This hope is satisfied, and this is indeed the way to approach coordinatization of Rmodule homomorphisms when R is not commutative. Now we come to a slight notational point. We could maintain the above notation for multiplication in RP throughout. This has two disadvantages: the practicalthat we would often be inserting the symbol which is
easy to overlook, and the theoreticalthat it makes the ring RIP look special (i.e., that for any "ordinary" ring we write multiplcation simply by juxtaposing elements, whereas in RIP we do not), whereas R°P is a perfectly
4.3 Matrix Representation of Homomorphisms
215
good ring, neither better nor worse than R itself. Thus, we adopt a second solution. Let op : R , RP be the function that is the identity ,n elements, i.e., op(t) = t for every t E R. Then we have op(sr) = op(r) op(s), where the multiplication on the righthand side, written as usual as juxtaposition, is the multiplication in R°P. This notation also has the advantage of reminding us that t E R, but op(t) E R°P. Note that if R is commutative then RIP = R and op is the identity, which in this case is a ring homomorphism. In fact, op : R + RP is a ring homomorphism if and only if R is commutative. In most applications of matrices, it is the case of commutative rings that is of primary importance. If you are just interested in the commutative case (as you may well be), we advise you to simply mentally (not physically) erase "op" whenever it appears, and you will have formulas that are perfectly legitimate for a commutative ring R. After this rather long introduction, we will now proceed to the formal mathematics of associating matrices with homomorphisms between free modules.
If R is a ring, M is a free Rmodule ofrank n,andB={vl,...,vn} is a basis of M, then we may write v E M as v = al vi +
+ anvn for unique a1, ... , an E R. This leads us to the definition of coordinates. Define
io:M + Mn,1(R°P)by op(al) op(an)
The n x 1 matrix [v]8 is called the coordinate matrix of v with respect to the basis B.
Suppose that B' = {vi, ... ,v;,} is another basis of M and define the matrix Ps,, E Mn(R°P) by the formula (3.1)
pt = [[v1]8'
[v2]8,
...
[vn]8']
That is, cold (Pg,) = [v3]13. The matrix P8 is called the change of basis
matrix from the basis B to the basis B'. Since vj = E 1 b,jv;, it follows that if B = 8' then PB = In. (3.1) Proposition. Let M be a free Rmodule of rank n, and let 8, 8', and B" be bases of M. Then
(1) for any v E M, [via' = Pt IV] B; (2) PB , = PB, PB; and (3) PB is invertible and (PB) ' = PB '. Proof. (1) Note that Mn,l(R°P) is an Rmodule where the operation of R on Mn,l (R°P) is given by r A = A op(r) for r E R, A E Mn,l (R°P). Then
216
Chapter 4. Linear Algebra
defined by ,b'(v) = [v]B,, is an Rmodule homomorphism, as is
0": M  MM,1(R°P),
defined by /"(v) = PB[v]B. To show that u" = 1/i' we need only show that they agree on elements of the basis B. If B = {vj, ... , then (vi]B = ei = Ei1 E M.,1(R) since vi = F,' 1 b;jv,. Thus, ti"(vi) = PB,ei = coli(PB,) =
='+G'(vi)
as required.
(2) For any v E M we have
(PB,PB,) [v]B = PX (PB'[v]B)
= PB %[v]B' = [v]B = PB [v]B. Therefore, PB;, PB, = P88,,.
(3) Take B" = B in part (2). Then
PB PB = PB = I,,. Thus, PB, is invertible and (PE) 1 =
PB,.
(3.2) Lemma. Let R be a ring and let M be a free Rmodule. If B' is any basis of M with n elements and P E GL(n, R°P) is any invertible matrix, then there is a basis B of M such that
P=Pg. Proof. Let B' = {vi, ... , vn} and suppose that P = (op(pii )]. Let v? _ 1 pijv;. Then B = {v1, ... , is easily checked to be a basis of M, and by construction, Pt = P.
,
Remark. Note that the choice of notation for the change of basis matrix P8 has been chosen so that the formula in Proposition 3.1 (2) is easy to remember. That is, a superscript and an adjacent (on the right) subscript that are equal cancel, as in
pt pt = PB,, . The same mnemonic device will be found to be useful in keeping track of superscripts and subscripts in Propositions 3.5 and 3.6.
4.3 Matrix Representation of Homomorphisms
217
(3.3) Definition. Let M and N be finite rank free Rmodules with bases
B = {vi, ... , v,n } and C = {w1, ... , w, ,J respectively. Fo, each f E HomR(M, N) define the matrix of f with respect to B, C, denoted [f]C, by
(3.2)
colj [ f ]C = [f (vj )]c
Thus, if f (vj)
for 1 < j:5 m.
aijwi for 1 < j < m, then [f]B = [op(aij)] E Mn,m(R°P)
If f E EndR(M) and B is a basis of M, then we will write [f]B in place of [f]8. (3.4) Remarks.
(1) Note that every matrix A E Mn,,n (RP) is the matrix [f) B for a unique f E HomR(M, N). Indeed, if B = {vi, ... , v,n} and C = {w1, ... , wn} are bases of M and N respectively, and if A = [op(aij )], then define f E
HomR(M, N) by f (vj) = E 1 aijwi. Such an f exists and is unique by Proposition 3.4.9; it is clear from the construction that A = [f]. Thus the mapping f '+ [f] B gives a bijection between HomR(M, N) and Mn,,n(R°P). (2) Suppose that R is a commutative ring. Then we already know (see Theorem 3.4.11) that HomR(M, N) is a free Rmodule of rank mn, as is the Rmodule Mn,,n(R); hence they are isomorphic as Rmodules. A choice of basis B for M and C for N provides an explicit isomorphism
HomR(M, N)  Mn,m(R),
defined by 4 (f) = (f). We leave it as an exercise for the reader to check that 4 ( f , ) = Eji where {E1}11 is the standard basis of Mn,m(R), while {fij}in1 is the basis of HomR(M, N) constructed 1
in the proof of Theorem 3.4.11.
Note that if 1M : M + M denotes the identity transformation and B, B' are two bases of M, then B
B
[lor]e = PB,
so that the change of basis matrix from the basis B to the basis B' is just the matrix of the identity homomorphism with respect to the matrix B on the domain and the basis B' on the range. (3.5) Proposition. With the above notation, if v E M then [.f(v)IC = [f]C[v]B
218
Chapter 4. Linear Algebra op(bi )
Proof. If [v)8 =
then op(bm)
f(v)=f E bjvj (j=1
M
_ >bjf(vj) j=1
 j=1 b' (atiwt) i=1 J
n
_ i=1
Therefore, If
m
(jaij
wi.
j=1
= [ E 1 op(biai,) ... F_j op(bjanj) I' = [flc [v] 8. 1
0
(3.6) Proposition. If M, N, and P are free Rmodules with bases B, C, and P are Rmodule homomorphisms, then
D, respectively, and f : M . N and g : N
[9°f11D1 = [91v[f1C
Proof. By Proposition 3.5, if v E M then [91v ([f]B[v1B) _ [91v[f(v))c
[9(f(v))ly _ [(9 ° f)(v)1v _ [9°f11[v1B.
Choosing v = vj = P element of the basis B so that [v] 8 = Ej1 E Mn,1(R°P),
we obtain [91v (colj((f]c)) = [91v ([f1c[vj]13)
= [9°f1[vj1B
=colj([9°fIS) for 1 < j < n. Applying Lemma 1.1 (6), we conclude that [9 ° f1v = [91D[f]C
as claimed.
4.3 Matrix Representation of Homomorphisms
219
(3.7) Remark. From this proposition we can see that matrix multiplication is associative. Let M, N, P, and Q be free Rmodules with ba.,es B, C, D,
and 6 respectively, and let f Rmodule homomorphisms. Then, by Proposition 3.6, L"ie (WJV[f]C) _ [hi9 ([9 ° f1D)
_ [h o (g o f )]B
_ [(hog)of]' [hog] [f 1 B
_ ([hlE [9]ai
[f J'3
By Remark 3.4 (1), every matrix is the matrix of a homomorphism, so associativity of matrix multiplication follows from the associativity of functional composition. (Actually, this proves associativity for matrices with entries
in RP, but then associativity for matrices with entries in R follows from the observation that R = (R°P)°P. Also observe that we used associativity in the proof of Proposition 3.1, but we did not use this proposition in the proof of Proposition 3.6, so our derivation here is legitimate.) (3.8) Corollary. Let M and N be free Rmodules of rank n. Let B be a basis of M and let C be a basis of N. Then a homomorphism f E HomR(M, N) is an isomorphism if and only if the matrix [fl' E Mn(R°P) is invertible. C
Proof. Suppose g = f 1 E HomR(N, M). Then, by Proposition 3.6, In = [1M]8 = [9 o f]BB = [918[f1C,
and similarly, In = [f1C[91B
Thus, (f] C
is an invertible matrix. The converse is left as an exercise.
0
(3.9) Corollary. Let R be a ring and let M be a free Rmodule of rank n. Then EndR(M) is isomorphic (as a ring) to Mn(R°Q). If R is a commutative ring, then this isomorphism is an isomorphism of Ralgebras. Proof. If B is a basis of M, let EndR(M)
Mn(R°P)
be defined by 4' (f) = [f]B. According to Proposition 3.6, 4iB is a ring homomorphism, while it is a bijective map by Remark 3.4. If R is commu0 tative, it is an Ralgebra isomorphism by Lemma 1.1 (3). (3.10) Remark. From Lemma 1.3 and Corollary 3.9 we immediately see that
if R is a commutative ring and M is a free Rmodule of finte rank, then the center of its endomorphism ring is
Chapter 4. Linear Algebra
220
C(EndR(M)) = R 1m. That is, a homomorphism f : M  M commutes with every other homomorphism g : M + M if and only if f =r1m for some r E R. For the remainder of this section we shall assume that the ring R is commutative.
(3.11) Proposition. Let R be a commutative ring, let M and N be free Rmodules, and let f E HomR(M, N). (1) If rank(M) < rank(N), then f is not surjective. (2) If rank(M) > rank(N), then f is not injective. (3) If rank(M) = rank(N) is finite and f is injective,
then N/ Im(f) is a
torsion Rmodule. (4) If rank(M) = rank(N) is finite and R is an integral domain, then f is
injective if and only if N/ Im(f) is a torsion Rmodule. Proof. (1) By the definition of rank (Definition 3.2.9), if f were surjective, then we would have rank(N) < rank(M). (2) If f were injective, N would contain a free submodule Im(f) of
rank(Im(f)) = rank(M) > rank(N), contradicting Corollary 2.32. (3) Let f be an injection, and let 7r : N p N/ Im(f) be the projection.
Suppose that N/ Im(f) is not torsion, and let n E N/ Im(f) be an element with Ann(n) _ (0). Let n E N with 7r (n) = n. Then Im(f) fl Rn = (0), and hence,
NDIm(f)®Rn, which is a free module of rank(M) + 1 = rank(N) + 1, contradicting Corollary 2.32. (4) Let R be an integral domain and assume that N/ Im(f) is a torsion module. Pick a basis {w1, ... , wn} of N. Since N/ Im(f) is a torsion mod
ule, there exists vi E M and ci # 0 E R with f (vi) = ciwi for 1 < i < n. Suppose V E M and f (v) = 0. Then the set {V, V1, ... , Vn}
is linearly dependent by Proposition 2.31, so let
an equation of linear dependence with not all of {a, a1, ... , an } equal to zero. Then
4.3 Matrix Representation of Homomorphisms
221
0=af(v) = f(av)
i=1
Since {w1, ... , wn } is a basis of N and R is an integral domain, it follows
that ai = 0 for all i. Hence av = 0 and thus v = 0 (similarly), and we conclude that f is an injection. (3.12) Remark. The assumption that R is an integral domain in Proposition
3.11 (4) is necessary. Let R = Zmn and set M = N = R. Let f : M . N be defined by f (v) = mv. Then Zmn/ Im(f) = Zn is a torsion Z,nnmodule, but f is not injective. The relationship between invertibility of homomorphisms and invertibility of matrices allows one to conclude that a homomorphimm between free Rmodules of the same finite rank is invertible if it has either a left or a right inverse.
(3.13) Proposition. Let M and N be free Rmodules of finite rank n, let B be a basis of M, and let C be a basis of N. If f E HomR(M, N), then the following are equivalent.
(1) f is an isomorphism. (2) f has a right inverse, i.e., there is a homomorphism g E HomR(N, M) such that f g = 1 N .
(3) f has a left inverse, i.e., there is a homomorphism h E HomR(N, M) such that hf = 1M. (4) f is a surjection.
s is an invertible matrix.
(5) (6)
[f]
(7)
(f )C has a left inverse.
(f ]
has a right inverse.
Proof. The equivalence of (5), (6), and (7) follows from Remark 2.18, while
the equivalence of (1), (2), and (3) to (5), (6), and (7), respectively, is a consequence of Corollary 3.8.
Now clearly (1) implies (4). On the other hand, assume that f is a surjection. Then there is a short exact sequence
0Ker(f),M
0.
Chapter 4. Linear Algebra
222
This sequence splits since N is free so that there is an Rmodule homomorphism g : N . M such that f9 = IN, i.e., g is a right inverse for f. Thus (4) implies (2), and the proof is complete. 0 (3.14) Remark. In Proposition 3.13 (4) it is not possible to replace surjective
by injective. It is true that if f has a left inverse, then f is injective, but the converse need not be true. For example, f : Z ' Z by f (x) = 2x is injective, but it is not left invertible. However, in case the ring R is a field, the converse is valid. This is the content of the following result.
(3.15) Proposition. Let F be a field and let M and N be vector spaces over F of dimension n. Then the following are equivalent.
(1) f is an isomorphism. (2) f is injective. (3) f is surjective. Proof. This is simply a restatement of Corollary 3.8.10.
0
(3.16) Proposition. Let M and N be free Rmodules with bases 8, 8' and C, C' respectively. If f : M N is an Rmodule homomorphism, then [f ]c and [ f )c, are related by the formula
[f15 = Pc[f1c (P8)1.
Proof. Since f = IN o f o 1M, Proposition 3.6 shows that (3.4)
(f1c = [lN1c [f1c(1M1g
But [1NJg, = PC, and [1M]B' = Pg' = (PS)', so Equation (3.3) follows from Equation (3.4).
0
We now give a determinantal criterion for the various properties of a homomorphism.
(3.17) Proposition. Let M and N be free Rmodules with rank(M) _ rank(N) finite. (1) f is surjective (and hence an isomorphism) if and only if in some (and, hence, in any) pair of bases B of M and C of N, det([ f ]B) is a unit of R.
(2) f is injective if and only if in some (and, hence, in any) pair of bases B of M and C of N, det([ f ]B) is not a zero divisor in R. Proof. (1) is immediate from Proposition 3.13 and Theorem 2.17 (2), while 0 part (2) follows from Corollary 2.27.
4.3 Matrix Representation of Homomorphisms
223
(3.18) Definition. (1)
Let R be a commutative ring. Matrices A, B E Mn,,n(R) are said to be equivalent if and only if there are invertible matrices P E GL(n, R) and Q E GL(m, R) such that
B = PAQ. Equivalence of matrices is an equivalence relation on Mn,,n(R). (2) If M and N are finite rank free Rmodules, then we will say that R
module homomorphisms f and g in HomR(M, N) are equivalent if there are invertible endomorphisms h1 E EndR(M) and h2 E EndR(N)
such that h2 f hi 1 = g. That is, f and g are equivalent if and only if there is a commutative diagram
M IM hll
+
N hs
9+ N
where the vertical maps are isomorphisms. Again, equivalence of homomorphisms is an equivalence relation on HomR(M, N). (3.19) Proposition.
(1) Two matrices A, B E Mn,m(R) are equivalent if and only if there are bases B, B' of a free module M of rank m and bases C, C' of a free module N of rank n such that A = [f] and B = [f]. That is, two c matrices are equivalent if and only if they represent the same Rmodule homomorphism with respect to different bases. (2) If M and N are free Rmodules of rank m and n respectively, then homomorphisms f and g E HomR(M, N) are equivalent if and only if there are bases B, B' of M and C, C' of N such that [f ]c = (gilc
l'.
That is, f is equivalent to g if and only if the two homomorphisms are represented by the same matrix with respect to appropriate bases. Proof. (1) Since every invertible matrix is a change of basis matrix (Lemma 3.2), the result is immediate from Proposition 3.16. (2) Suppose that [f]c = [g]c,. Then Equation (3.3) gives (3.5)
[f ]c = [91 c' = l c, [9]c (PB) '
The matrices Pg and PC are invertible so that we may write (by Corollary 3.9) Pg = [h1]5 and [h2]c where h1 E EndR(M) and h2 E EndR(N) are invertible. Thus Equation (3.5) gives
If)" = [h2]c[9]c ([h1]g)1 = [h29hi 1]c c
224
Chapter 4. Linear Algebra
Hence, f = hsghi 1 and f and g are equivalent. The converse statement is left as an exercise.
0
Using the invariant factor theorem for submodules (Theorem 3.6.23), it is possible to explicitly describe the equivalence classes under the equivalence relations of equivalence of homomorphisms and equivalence of ma
trices if one restricts the ring R to be a PID. This is the content of the following result.
(3.20) Proposition. Let R be a PID and let f : M N be an Rmodule homomorphism between a free Rmodule M of rank m and a free Rmodule
N of rank n. Then there is a basis B = {vl, ... , v,,,} of M, a basis C = {wl, ... , of N, and nonzero elements sl, ... , ar of R, where r = rank Im(f ), such that s; I si+1 for 1 < i < r  1 and such that I f (vi) = siwi f (Vi) =0
if l < i < r, if i > r.
That is, the matrix of f with respect to the bases B and C is [fJc=[Or
01
where Dr = diag(81, ... , sr). Proof. By the invariant factor theorem for submodules (Theorem 3.6.23), of N and elements al, ... ,sr E R such there is a basis C = {wl, ...
that si 1 8i+1 for 1 < i < r  1 and {81w1i ... , srwr} is a basis for the submodule Im(f) C N. Now choose any subset {v1, ... , v,) C M such that f (vi) = aiwi for 1:5 i < r. By Proposition 3.8.8, Ker(f) is a submodule of M of rank m  r. Thus, we may choose a basis {vr+1, ... , v,,,) of Ker(f).
Claim. B = {vi, ... , vr, Vr+1,
,
v,n} C M is a basis of M.
To verify the claim, suppose that v E M. Then f (v) E Im(f ), so we may write f(v) _ >2 aif(vi) i=1
f(aivi)
_ i=1
Therefore, f (v we may write
aivi) = 0 so that v r
aivi E Ker(f ), and hence,
m
V  E aivi = E aivi. i=1
i=r+1
It follows that 8 generates M as an Rmodule.
4.3 Matrix Representation of Homomorphisms
To check linear independence, suppose that m
225
Ein 1 aivi = 0. Then
r
0 = >aif(vi) _ >ai(siwi) :=1
{=1
Since {s1w1i ... , srwr} is linearly independent, this implies that ai = 0 for
1 < i < r. But then
m
aivi = 0, ir+1
is a basis of Ker(f ), we conclude that ai = 0 for and since {vr+1, ... , all i, and the claim is verified.
It is clear from the construction that fflcs
=
Dr
0
0
o1
where Dr = diag(s1i ... , sr). This completes the proof.
0
(3.21) Remark. In the case where the ring R is a field, the invariant factors are all 1. Therefore, if f : M  N is a linear transformation between finitedimensional vector spaces, then there is a basis B of M and a basis C of N
such that the matrix of f is 1f1B
=
10r
0,
The number r is the dimension (= rank) of the subspace Im(f).
(3.22) Corollary. Let R be a PID and let M and N be free Rmodules of rank m and n respectively. Then homomorphisms f and g E HomR(M, N) are equivalent if and only if the subspaces Im(f) and Im(g) have the same invariant factors as submodules of N. In particular, if R is a field, then f and g are equivalent if and only if rank(Im(f )) = rank(Im(g)).
0
Proof. Exercise.
If M = N, then Proposition 3.16 becomes the following result:
(3.23) Proposition. Let f E EndR(M) and let B, B' be two bases for the free Rmodule M. Then [f] B' = PB [f]B (PB,)1.
Chapter 4. Linear Algebra
226
Proof.
Proposition 3.23 applies to give a result analogous to Proposition 3.19 for similarity of matrices in M,,(R). Recall (Definition 1.7) that two matrices A and B in M,,(R) are similar if there is a matrix S E GL(n, R) such that B = S' AS. For homomorphisms, the definition is the following: (3.24) Definition. Let R be a commuative ring and let M be a free Rmodule of rank n. If j r, g E EndR(M), then we say that f and g are similar if there is an invertible homomorphism h E EndR(M) such that g = h' f h.
In this situation, Proposition 3.19 becomes the following result: (3.25) Corollary. (1) Two matrices A, B E Mn (R) are similar if and only if there are bases
B and B' of a free Rmodule M of rank n and f E HomR(M) such that A = [f)13 and B = if ]s . That is, two n x n matrices are similar if and only if they represent the same Rmodule homomorphism with respect to different bases. (2) Let M be a free Rmodule of rank n and let f, g E EndR(M) be endomorphisms. Then f and g are similar if and only if there are bases B and B' of M such that [f]L; = [g)s'
That is, f is similar to g if and only if the two homomorphisms are represented by the same matrix with respect to appropriate bases. Proof. Exercise.
(3.26) Remark. Let R be a commutative ring and T a set. A function 0 : Mn(R)  T will be called a class function if O(A) =,O(B) whenever A and B are similar matrices. If M is a free Rmodule of rank n, then the class function 0 naturally yields a function 4' : EndR(M)  T defined by 4'(f) = m([fIB) where B is a basis of M. According to Corollary 3.25, the definition of m is independent of the choice of basis of M because 0 is a class function. The
most important class functions that we have met so far are the trace and the determinant (Lemma 1.6 (2) and Corollary 2.12). Thus, the trace and the determinant can be defined for any endomorphism of a free Rmodule of finite rank. We formally record this observation. (3.27) Proposition. Let R be a commutative ring and let Al be a finite rank free Rmodule.
4.3 Matrix Representation of Homomorphisms (1)
227
There is an Rmodule homomorphism Tr : EndR(M) + R defined by
Tr(f) = T VI is) where B is any basis of M. Tr(f) will be called the trace of the homomorphism f; it is independent of the choice of basis B. (2) There is a multiplicative function det : EndR(M)  R defined by
det(f) = det([fJB) where B is any basis of M. det(f) will be called the determinant of the homomorphism f ; it is independent of the choice of basis B. Proof.
Note that multiplicativity of the determinant means det(fg) = det(f) det(g).
Since det(1) = 1, it follows that f is invertible if and only if det(f) is a unit in R. Since similar matrices represent the same endomorphism with respect to different bases, one goal of linear algebra is to find a matrix B similar to a given matrix A such that B is as simple as possible. This is equivalent (by Corollary 3.25) to finding a basis of a free Rmodule so that the matrix [ f ]B of a given homomorphism f is as simple as possible. When R is a field, this is the subject of canonical form theory that will be developed in detail in the next section. For now we will only indicate the relationship between direct sum decompositions of free Rmodules and decompositions of matrices. (3.28) Proposition. Let R be a commutative ring, and let M1, M2, N1, and N2 be finite rank free Rmodules. If Bi is a basis of Mi and Ci is a basis of Ni (i = 1, 2), then let B1 U B2 and C1 U C2 be the natural bases of Ml ® M2 and N1 ® N2 respectively (see Example 3.4.6 (7)). If fi E HomR(M1, Ni) for i = 1, 2, then fl ® f2 E HomR(M1 ® M2, N1 ® N2) and UB2 = [f1JB' ® [f1® f2 JcBlUc,
[f2JB2
2.
Proof. Exercise.
We now specialize to the case of endomorphisms.
(3.29) Definition. Let M be an Rmodule and let f E EndR(M). A submod
ule N C M is said to be invariant under f (or an invariant submodule of f) if f (x) E N whenever x E N.
228
Chapter 4. Linear Algebra
(3.30) Proposition. Let R be a commutative ring, let M be a free R module
of rank m, and let f E HomR(M). If B = {v1, ... , vm } is a basis of M then the matrix [f] B has the block form [fit;
_
AB
0D
where A E Mr(R) if and only if the submodule N = (v1, ... , v,.) is an invariant submodule of f. Proof. If if JB = [tip) then the block form means that tip = 0 for r + 1 < i
t,.,vi =
t3vi E N.
0
Since N is generated by v1, ... , V,., the result follows.
(3.31) Remark. As a special case of this result, a matrix [1 [B is upper triangular if and only if the submodule (v1, ... vk) (where B = {vj, ... ,v,,,}) is invariant under f for every k (1 < k < m).
In Proposition 3.30, if the block B = 0, then not only is (vi, ...
, vk)
an invariant submodule, but the complementary submodule (vk+1, ... , v,,,) is also invariant under f. From this observation, extended to an arbitrary number of blocks, we conclude:
(3.32) Proposition. Let M be a free Rmodule of rank m, let f E EndR(M), and let A = [fie E Mm(R). Then A is similar to a block diagonal matrix
B = Al ® ® Ak where Ai E Al, (R) (rl +  + rk = n) if and only if there are free submodules All, ... , Mk of M such that
(1) M; is an invariant submodule off which is free of rank ri, and (2) M All ED ...ED Mk. Proof
0
The case of this result when r, = 1 for all i is of particular importance. In this case we are asking when A is similar to a diagonal matrix. To state the result in the manner we wish, it is convenient to make the following definition.
(3.33) Definition. If M is a free Rmodule of rank m and f E EndR(M), then a nonzero x E M is called an eigenvector off if the cyclic submodule (x) of M is invariant under f. That is, x 0 E M is an eigenvector of Al if and only if f (x) = ax for some a E R. The element a E R is called
4.3 Matrix Representation of Homomorphisms
229
an eigenvalue of f . The eigenniodule (or eigenspace) off corresponding to the eigenvalue a is the submodule Ker(f  a1M). If A E Mn(R) then by an eigenvalue or eigenvector of A, we mean an eigenvalue or eigenvector of the Rmodule homomorphism
TA : M.,i(R)  Mn.1(R) defined by TA(v) = Av. We shall usually identify Mn,1(R) with R' via the standard basis {Eti1 : 1 < i < n} and speak of TA as a map from R" to R". In practice, in studying endomorphisms of a free Rmodule, eigenvalues and eigenvectors play a key role (for the matrix of f depends on a choice of basis, while eigenvalues and eigenvectors are intrinsically defined). We shall consider them further in Section 4.4.
(3.34) Corollary. Let M be a free Rmodule of rank m, let f E EndR(M),
and let A = [f]g E Mn(R). Then A is similar to a diagonal matrix diag(ai, ... ,a.) if and only if there is a basis of M consisting of eigenvectors of A.
0
Proof.
(3.35) Definition. A matrix which is similar to a diagonal matrix is said to be diagonalisable. An endomorphism f E EndR(M) is diagonalizable if M has a basis B such that [f]13 is a diagonal matrix. A set S = { fi }iE j of endomorphisms of M is said to be simultaneously diagonalizable if M has a basis B such that [fi]s is diagonal for all i E I. The concept of diagonalizability of matrices with entries in a field will
be studied in some detail in Section 4.4. For now we will conclude this section with the following result, which we shall need later.
(3.36) Theorem. Let R be a PID, let M be a free Rmodule of finite rank
n, and let S = { fi : M  M}$EJ be a set of diagonalizable Rmodule endomorphisms. Then the set S is simultaneously diagonalizable if and only
if S consists of commuting endomorphisms, i.e., there is a basis B of M with [fi]n diagonal for all i E I if and only if fi fi = f, fi for all i, j.
Proof. For simplicity, we assume that we are dealing with a pair of Rmodule endomorphisms {f, g}; the general case is no more difficult. First suppose that B = {v1, ... , vn } is a basis of M in which both [ f ]8 and [g]j; are diagonal. Then f (vi) = \;v; and g(vi) = µivi for 1 < i < n. Then f(9(vi)) = f(µivi) = µi9(vi) = /1i'\ivi
= '\wivi = Ai9(vi) = 9(A1vi)
= 9(f (vi)), so f g and g f agree on a basis of M and hence are equal.
230
Chapter 4. Linear Algebra
Conversely, suppose that f g = g f and let Al, ... , Ak be the distinct eigenvalues of f, and let A,, ... , ,ut be the distinct eigenvalues of g. Let
Mi = Ker(f  Ailm) Nj = Ker(g  µj I t j)
(3.8)
(3.9)
(1 < i < k) (1 < j < Q).
Then the hypothesis that f and g are diagonalizable implies that (3.10)
MI®...(D Mk
and (3.11)
M'NI(D ...ED Nt.
First, we observe that Mi is ginvariant and Nj is finvariant. To see this, suppose that v E Mi. Then f(g(v)) = Of M) = g(Aiv) = Aig(v),
i.e., g(v) E Ker(f  Ai1M) = Mi. The argument that Nj is finvariant is the same.
Claim. Mi=®j=1(MinN3) fort deg(fi(X )) = dim(V) = deg(cT(X )). i=1
(3)
The following are equivalent: (a) VT is cyclic. (b) deg(mT(X)) = dim(V).
(c) MAX) = CT(X)Proof. (1) Suppose that VT = Rv. Then the map q : R
VT defined by
q(T)(v)
R/ Ker(i) as Fmodules. But Ker(rl) _ (f (X)), and as Fmodules, R/(f(X)) has a basis {1, X, ... ,Xn1} where n = is surjective and vT
deg(f (X)). Thus, dimF(V) = n = deg(f (X)). (2) and (3) are immediate from (1) and the definitions.
0
(4.12) Definition. Let f (X) = Xn + + a1X + a0 E F[X] + be a monic polynomial. Then the companion matrix C(f(X)) E Mn(F) of f (X) is the n x n matrix an_1Xn1
4.4 Canonical Form Theory
(4.13)
C(f M) =
0
0
0
0
ao
1
0
0
al
0
1
0
0 0
I0
0 0
1
0
0
1
0
.
237
a2
an1 °n2 J
(4.13) Examples.
(1) For each AEF,C(XA)=[a]EM1(F). (2) diag(al, ... an) = ®i 1C(X  a,). (3) C((2 + 1) = (4)
r0 Ol l
If A = C(X  a) ®C(X2  1), then a 0
A= 0
0
0
0
1
1
0
(4.14) Proposition. Let f (X) E F[X] be a monk polynomial of degree n,
and let T : Fn  F" be defined by multiplication by A = C(f(X)), i.e., T(v) = Av where we have identified F" with Mn,1(F). Then mT(X) _
f(X). Proof. Let ej = col,(In). Then from the definition of A = C(f (X)) (Equation (4.13)), we see that T(e1) = e2, T(e2) = e3, ..., T(en1) = en. There
fore, T1(e1) = e,.+1 for 0 < r < n  1 so that {e1,T(e1), ... ,T"'(el)} is a basis of Fn and hence (F")T is a cyclic Rmodule generated by e1. Thus, by Lemma 4.11, deg(mT(X)) = n and
(mT(X)) = Ann((F")T) = Ann(e1). But T"(e1) = aoe1  a1T(e1) 
an1Tn1(e1),
i.e., T"(el)+an1T"1(el)+...+a1T(e1)+aoe1 =0.
Therefore, f(T)(e1) = 0 so that f(X) E Ann(e1) = (mT(X)). But deg(f (X )) = deg(mT(X)) and both polynomials are monic, so f (X) = mT(X).
238
Chapter 4. Linear Algebra
(4.15) Corollary. In the situation of Proposition 4.14, let
8 = {v,T(v), ... ,T' '(v)} where v = el = coll (I"). Then B is a basis of F" and [T]B = C(f (X )). Prof. This is clear from the calculations in the proof of Proposition 4.14. 0 (4.16) Corollary. Let V be any finitedimensional vector space over the field
F, let T E EndF(V), and suppose that the Rmodule VT is cyclic with Ann(VT) = (f (X )) . Then there is a basis 8 of V such that [T]13 = C(f (X)).
Proof. If VT = Rv then we may take
8 = {v,T(v), ... ,T"1(v)}
0
where n = dim(V).
(4.17) Theorem. (Rational canonical form) Let V be a vector space of dimension n over a field F and let T E EndF(V) be a linear transformation. If {f1(X), ... , fk(X)} is the set of invariant factors of the F[X]module VT, then V has a basis B such that (4.14)
[T)B = C(f1(X )) ® C(f2(X )) ® ... ED C(fk (X)).
Proof. Let VT ^_' Rv1 ® ... ® Rvk where R = F[XI and Ann(vi) = (fi(X))
and where fi(X) I fi+1(X) for 1 < i < k. Let deg(f;) = ni. Then Bi = {vi, T(vi), ... ,T1,1(vi)} is a basis of the cyclic submodule Rvi. Since submodules of VT are precisely the Tinvariant subspaces of V, it follows that TAR,,; E EndF(Rvi) and Corollary 4.16 applies to give (4.15)
(TIRvj , =
C(fi(X))
By Equation (4.12), ni + + nk = n, and hence, B = U;= I is a basis of V and Proposition 3.32 and Equation (4.15) apply to give Equation (4.14).
0 (4.18) Corollary. Two linear transformations T1 and T2 on V have the same rational canonical form if and only if they are similar.
Proof. Two linear transformations T1 and T2 have the same rational canonical form if and only if they have the same invariant factors, which occurs if and only if the Rmodules VT, and VT, are isomorphic. Now apply Theorem 4.2.
0
4.4 Canonical Form Theory
239
(4.19) Corollary. Every matrix A E Mn(F) is similar to a unique matrix in rational canonical form.
Proof. Regard A E Mn(F) as defining a linear transformation TA Fn . Fn by TA(v) = Av. Then A is similar to B E Mn(F) as matrices if and only if TA is similar to TB as elements of EndF(Fn). Thus we may apply Theorem 4.17 and Corollary 4.18. We now pause in our general development in order to see how to compute cT(X) and to prove a famous result. We will need the following simple lemma:
(4.20) Lemma. Let F be a field and let f (X) E FIX) be a monic polynomial
of degree n. Then the matrix XIn  C(f (X)) E Mn(F[XJ) and (4.16)
det (XIn  C(f (X))) = f(X).
Proof. Let f (X) = Xn + + a1X + ao E FIX]. The proof + is by induction on n = deg(f). If n = 1, the result is clear. Now suppose that n > 1, and compute the determinant of Equation (4.16) by cofactor an_1Xn1
expansion along the first row; applying the induction hypothesis to the first summand.
det(XIn  C(f (X))) = det 1
X
0
1 0
X 1
0 0
0 0
X
. .. . .. . ..
. .
0 0 0
0 0 0
ao a1
a2
1 X
.
...
0
1 ...
0
an2
1 X+an_1
0 0 0
a1
a2
= X det 0
1
0
0
X an2 1 X +an_1
1 X
...
0
1
..
0 0
0 0
...
+ a0(_1)n+1 det
0 0
0 0
1 X
= X (Xn1 + an_1Xn2 +... + al) +ao(1)n+l(_1)n1
= Xn+an_1Xn1 +...+a1X +ao
0
1
240
Chapter 4. Linear Algebra
= f(X), and the lemma is proved.
(4.21) Definition. If A E Mn(F) then we will denote the polynomial det(XIn  A) E F[X] by CA(X) and we will call cA(X) the characteristic polynomial of the matrix A.
We will prove that the characteristic polynomial of a linear transformation T (as defined in Definition 4.6 (3)) is the characteristic polynomial of any matrix representation of T.
(4.22) Lemma If A and B E Mn(F) are similar, then cA(X) = CB(X)
Proof. Suppose that B = P'AP for some P E GL(n, F). Then det P 0 0 and (det P)1 = det P1. Hence,
CB(X) = det(XI  B) = det(XIn  P'AP) = det(P1(XIn  A)P) = (detP1)(det(XIn  A))(detP) = det(XIn  A) = CA(X) 0 (4.23) Proposition. Let V be a finitedimensional vector space over a field F and let T E EndF(V). If B is any basis of V, then (4.17)
cT(X) = c(Tie(X) = det(XIn  (T]B).
Proof. By Lemma 4.22, if Equation (4.17) is true for one basis, it is true for any basis. Thus, we may choose the basis B so that (T] B is in rational canonical form, i.e., (4.18)
(T]B = ®C(fi(X)) i=1
where f1(X), ... , fk(X) are the invariant factors of T. If deg(fi(X)) = n then Equation (4.18) gives k
(4.19)
XIn  (T]B = ®(XIn,  C(fi(X ))) i=1
Equation (4.19) and Theorem 2.11 imply
4.4 Canonical Form Theory
241
clTl8(X) = det(XI  [T18) k
det(XI,,, C(fi(X)))
= i1 k
_ fj fi (X)
by Lemma 4.20
i=1
= CT(X),
0
which gives Equation (4.17).
(4.24) Corollary. (CayleyHamilton theorem) Let T E EndF(V) be any linear transformation on a finitedimensional vector space V and let B be any basis of V. Let A = [TJ8 E F[XJ. Then CA(T) = 0.
Proof. By Proposition 4.23, cA(X) = cT(X) and mT(X) I cT(X). Since 0 mA(T) = 0, it follows that cT(T) = 0, i.e., cA(T) = 0. (4.25) Remark. The CayleyHamilton theorem is often phrased as A linear transformation satisfies its characteristic polynomial. From our perspective, the fact that cT(T) = 0 is a triviality, but it is a nontrivial result that CT(X) = cA(X) where A = [TJ8. However, there is an alternate approach in which the characteristic polynomial of a linear transformation is defined to be CA(X) where A is some matrix representation of T. From this perspective, the CayleyHamilton theorem becomes a nontrivial result. It is worth pointing out that the CayleyHamilton theorem is valid for matrices with entries in any commutative ring; of course, the invariant factor theory is not valid for general rings so a different proof is needed (see Exercise 56). In fact, we shall sketch a second proof of the CayleyHamilton theorem in the exercises which is valid for any commutative ring R. From the point of view of the current section, the utility of Proposition 4.23 and Corollary 4.24 is that we have an independent method of calculating the characteristic polynomial. Further techniques for computing the invariant factors of a given linear transformation will be presented in Chapter 5.
(4.26) Example. Let V be an ndimensional vector space over the field F,
and if A E F, define a linear transformation T : V  V by T(v) = av for all v E V. Then, considering the F[XJmodule VT, we have Xv = Av,
i.e., (X  A)v = 0 for every v E V. Thus X  A = mT(X) since X  A is the monic polynomial of lowest possible degree (namely, degree 1) with
MT(T) = 0. Then cT(X) = (X  A)' since deg(or(X)) = n and the only prime factor of cT(X) is (X  A) by Corollary 4.10 (3). Then the rational canonical form of T is
242
Chapter 4. Linear Algebra n
®C(XA)=AIn. i=1
Of course, this is the matrix of T in any basis of V. Also note that VT 25
Rv1 ® ® Rvn has rank n over R with each cyclic submodule Rv, 25 R/(X  A) having dimension 1 over the field F.
(4.27) Example. Let B = { v1i ... , vn } be a basis of the ndimensional
vector space V, and let T : V . V be defined by T(vi) = A,v, where A, E F. Assume that the A, are all distinct, i.e., a, # A3 for i 0 j. Then each subspace (v,) is a Tinvariant subspace and hence a submodule of the F[X )module VT. Therefore,
VT =Rv,e...®Rvn where Ann(vi) = (X  A,). Note that
me(Rv,) = Ann(vi) = (X  A,), so Proposition 3.7.21 implies that MT(X) = me(VT)
= lcm{me(Rvl), ... , me(Rvn)} n
=
fl(X
 Ai)
i=1
= f(X) Also by Proposition 3.7.21, we see that cT(X) = f (X). Therefore MT(X) _ cT(X) and Lemma 4.11 (3) shows that the Rmodule VT is cyclic with annihilator (f (X)). Thus the rational canonical form of T is 0 1
0 0
0
1
0 0
0 0
... ...
0
0
0 0
0 0
a0 a1 a2
1
0
an_2
0
1
an_,
[T)eo = C(f(X)) =
where f (X) = X' + an 1 X n 1 + appropriately.
...
+ a 1 X + ao and the basis Bo is chosen
This example actually illustrates a defect of the rational canonical form. Note that [T)5 = diag(A1, A2, ... , An), which is a diagonal matrix. By comparison, we see that [T)13 is much simpler than [T] and it reflects
the geometry of the linear transformation much more clearly. Our next goal is to find a canonical form that is as "simple" as possible, the Jordan
4.4 Canonical Form Theory
243
canonical form. When a transformation has a diagonal matrix in some basis, this will indeed be its Jordan canonical form. This special case is important enough to investigate first. (4.28) Definition.
(1) A linear transformation T : V * V is diagonalizable if V has a basis such that 171E is a diagonal matrix.
(2) A matrix A E Mn(F) is diagonalizable if it is similar to a diagonal matrix.
(4.29) Remark. Recall that we have already introduced the concept of diagonalizability in Definition 3.35. Corollary 3.34 states that T is diagonalizable if and only if V posseses a basis of eigenvectors of T. Recall that v 0 E V is an eigenvector of T if the subspace (v) is Tinvariant, i.e., T(v) = Av for
some A E F. The element A E F is an eigenvalue of T. We will consider criteria for diagonalizability of a linear transformation based on properties of the invariant factors.
(4.30) Theorem. Let T : V  V be a linear transformation. Then T is diagonalizable if and only if mT(X) is a product of distinct linear factors, i. e., t
mT(X) = [J(X  Ai) i=1
where A1i ... , At are distinct elements of the field F.
Proof. Suppose that T is diagonalizable. Then there is a basis B of V such that [T]B = diag(a1i ... , an). By reordering the basis B, if necessary, we can assume that the diagonal entries that are equal are grouped together. That is, [T]B = ® Ailn i=1
where n = n1 +
+ nt and the Ai are distinct. If
B = {v11, ... t V1n,, V21, ...
, V2n2, .
. .
, vtl, ... , vine },
then let Bi = {vi1, ... , vin, } and let V = (B,). Then T(v) = Aiv for all v E Vi, so Vi is a Tinvariant subspace of V and hence an F[X]submodule of VT. from Example 4.26, we see that me(V) = X  A,, and, as in Example 4.27,
mT(X) = me(VT) = H(X  A1) i=1
as claimed.
244
Chapter 4. Linear Algebra
Conversely, suppose that mT(X) = rj;=1(X  Ai ), where the Ai are distinct. Since the X ,\i are distinct irreducible polynomials, Theorem 3.7.13 applied to the torsion F[X]module provides a direct sum decomposition (4.20)
VT
ED
ED
t
where Ann(Vi) = (X  Ai). In other words, Vi is a Tinvariant subspace of V and Ti = TIy, satisfies T,  Ai = 0, i.e., T(v) = Aiv for all v E V,. Then, by Example 4.26, if Vi has a basis Bi = (vi1, ... , vi,,), then [T;]8, = Ail,,,, and if B = U;=1Bi, then B is a basis of V by Equation (4.20) and [T]8 = so T is diagonalizable. (4.31) Corollary. Let T E EndF(V) be diagonalizable. Then the exponent of (X  Ai) in the characteristic polynomial cT(X) is equal to the number of times that Ai appears on the diagonal in any diagonal matrix (T]8.
Proof. If Ai appears ni times on the diagonal in a diagonal matrix representation of T, then
n, = dimF Ker(T  Ail,). This number depends only on T and not on any particular diagonalization of T. Now suppose that (T]8 = ®;=1Ai1,,,, let Vi = Ker(T  ally), and let Ti = T I v, as in the proof of Theorem 4.30. Then by Proposition 3.7.21 cT(X) = co(T) = co(T1)
co(Tt)
=CT'(X)...CT,(X) = (X  A1)"' ... (X  At)"' as claimed.
Since, by Proposition 4.23, we have an independent method for calculating the characteristic polynomial cT(X), the following result is a useful sufficient (but not necessary) criterion for diagonalizability.
(4.32) Corollary. Let V be an ndimensional vector space and let T E EndF(V) be a linear transformation. If the characteristic polynomial cT(X)
is a product of distinct linear factors, then MT(X) = cT(X) and hence T is diagonalizable.
Proof. Suppose that cT(X) = rj; 1(X ,\i) where the Ai are distinct. Since every irreducible factor of cT(X) is also a factor of MTV), it follows that CT(X) divides mT(X). But since mT(X) always divides cT(X) (Corollary 4.10), it follows that mT(X) = cT(X ). The diagonalizability of T then follows from Theorem 4.30.
4.4 Canonical Form Theory
245
(4.33) Remark. It is interesting to determine the cyclic decomposition of VT when T is diagonalizable. By Theorem 4.30, we have
mT(X) _ (X  A1) ... (X  At) and
cT(X) = (X  A1)" ... (X  At)nk
+ nt. If k = where the A, are distinct and n = dim(V) = nl + Rvl ® ® Rvk, and the inmax{nl, ... , nt} then rank(VT) = k, VT variant factors of the torsion Rmodule VT are the polynomials
fk+1_i(X) = (X  A)e(+,nt) ... (X  At)e(i,nt) where 1
IE(i'j) =
0
for
1 < i < k,
ifi<j, if i > j.
The following special case of Theorem 4.30 will be useful to us later.
(4.34) Corollary. Suppose that V is a finitedimensional vector space over
a field F and let T : V + V be a linear transformation such that TIC = 1y. Suppose that F is a field in which the equation zk = 1 has k distinct solutions. Then T is diagonalizable.
Then X k  1 = Proof. Let the solutions of zk 1 be 1 I [J k (X  () is a product of distinct linear factors. By hypothesis, Tk ly =o0, so T satisfies the polynomial equation Xk  1 = 0, and hence mT(X) divides Xk 1. But then mT(X) is also a product of distinct linear factors, and hence T is diagonalizable. (4.35) Remark. Note that the hypothesis on the field F is certainly satisfied for any k if the field F is the field C of complex numbers. From Theorem 4.30, we see that there are essentially two reasons why a linear transformation may fail to be diagonalizable. The first is that mT(X)
may factor into linear factors, but the factors may fail to be distinct; the second is that mT(X) may have an irreducible factor that is not linear. For example, consider the linear transformations T. : F2 by multiplication by the matrices
Al=[0 0]
r
and
F2, which are given 11
A2=[01 p1.
Note that mT, (X) = X2, so Tl illustrates the first problem, while mT, (X) _ X2 + 1. Then if F = R, the real numbers, X2+1 is irreducible, so T2 provides an example of the second problem. Of course, if F is algebraically closed (and in particular if F = C), then the second problem never arises.
246
Chapter 4. Linear Algebra
We shall concentrate our attention on the first problem and deal with the second one later. The approach will be via the primary decomposition theorem for finitely generated torsion modules over a PID (Theorems 3.7.12 and 3.7.13). We will begin by concentrating our attention on a single primary cyclic Rmodule.
(4.36) Definition. Let A be in the field F and let n E N. An n x n Jordan block with value A is the matrix A
1
0
0
A
1
0 0
0
0
0
0
(4.21)
Note that Ja,n = AIn + Hn where n1
Ei,i+1 E Mn(F)
Hn =
i1
That is, Hn has a 1 directly above each diagonal element and 0 elsewhere. Calculation shows that nk
(Jan  Aln)k = Hn =
Ei i+k
0
for
1 0, so 0(y) 0 and Ker(4) _ (0). (6.13) Theorem. Let V be a finitedimensional inner product space. Then for every T E EndF(V) there exists a unique T' E EndF(V) such that
(Tv:w)=(v:T'w) for all v, w E V. T' is called the adjoint of T. Proof. Let w E V. Then h,,, : V + F defined by h,,,(v) = (Tv : w) is an element of the dual space V. Thus (by Remark 6.12), there exists a unique w E V such that
(Tv:w)=(v:w) for all v E V. Let T'(w) = w. We leave it as an exercise to verify that T' E EndF(V). 0 (6.14) Lemma. Let V be an inner product space and let S, T E EndF(V).
Then (ST)' = T'S'.
0
Proof. Exercise.
(6.15) Lemma. Let V be a finitedimensional inner product space and let 8 be an orthonormal basis of V. If T : V + V is a linear transformation and (T)e = A = [ai3], then
[T*]e=A'=A`
274
Chapter 4. Linear Algebra
Proof. Let B = {vi, ...
If T'(vi) = F,k=lbkivk, then, according to
Lemma 6.8,
bji = (T* (vi) : vj) = (vj : TO (vi))
= (T(vj) : vi) n
_
\u akj Vk : k=1
Vi
ll
= Qij.
Thus, [T')B = T, as required.
0
(6.16) Definition. Let V be an inner product space, and let T : V  V be a linear transformation.
(1) T is normal if TT' = T'T, i.e., if T commutes with its adjoint. (2) T is selfadjoint if T = T', i.e., if T is its own adjoint. (3) T is unitary i f T' = T ' .
LetAEMn(F) be an n x n matrix. (1') A is normal if AN = A' A. (2') A is selfadjoint if A = N. (3') A is unitary if AN = I,,. (6.17) Remarks. (1) (2)
(3)
If T is selfadjoint or unitary, then it is normal. If F = C, then a selfadjoint linear transformation (or matrix) is called Hermitian, while if F = R, then a selfadjoint transformation (or matrix) is called symmetric. If F = R, then a unitary transformation (or matrix) is called orthogonal. Lemma 6.15 shows that the concept of normal is essentially the same for transformations on finitedimensional vector spaces and for matrices. A similar comment applies for selfadjoint and unitary.
The importance of unitary transformations arises because of the following geometric property which characterizes them. (6.18) Proposition. Let T : V V be a linear transformation on the finitedimensional inner product space V. The following are equivalent:
(1) T is unitary. (2) (Tu : Tv) = (u: v) for all u, v E V. (3) IITvIJ = Jlvi) for all v E V.
4.6 Inner Product Spaces and Normal Linear Transformations
275
Proof. (1) =: (2). Let u, v E V. Then
(Tu: Tv) = (u : T'Tv) = (u : T'Tv)
=(u:v). (2) = (1). (u : v) _ (Tu : Tv) _ (u : T'Tv), so
(u:T'Tvv)=0
for all
u,vEV.
Taking u = T'Tv  v shows that (u : u) = 0. Thus u = 0, i.e.,
T'Tv = v for all v E V. Therefore, T'T = 1v, and since V is finite dimensional, this shows that
T' = T'. (2) * (3). Recall that IjTvjI2 = (Tv : Tv). (3) . (2). Let u, v E V. Then IIT(u + v)HH2 = (T(u + v) : T(u + v))
= (Tu : Tu) + (Tu : Tv) + (Tv : Tu) + (Tv : Tv) = (u : u) + 2Re ((Tu : Tv)) + (v : v), while similarly
(u+v:v+v)=(u: u) + 2Re ((u: v)) + (v: v). Thus, we conclude that
Re((Tu : Tv)) = Re(u : v). Applying the same argument to u + iv, we obtain
Re ((Tu : iTv)) = Re ((Tu : T(iv)) = Re(u : iv). But it is easy to check that Re((x : iy)) = Im((x : y)) for any x, y E V, so we have
(Tu: Tv) = (u: v)
forallu,vE V.
0
The following result collects some useful properties of normal transformations.
(6.19) Lemma. Let V be a finitedimensional inner product space and let T : V + V be a normal transformation.
276
(1) (2)
Chapter 4. Linear Algebra
If f(X) E FIX) then f(T) is normal.
IITvII=IIT'vIIforallvEV.
(3) Ker T = (Im T)1. (4) If T2v = 0 then Tv = 0. (5) v E V is an eigenvector for T with eigenvalue A if and only if v is an eigenvector for T' with eigenvalue A.
Proof. (1) This follows immediately from the fact that (aT")' = a(T')" and the definition of normality. (2)
IITvII2
= (Tv : Tv) = (v : T'Tv)
=(v:TT'v)=(T'v:T'v) = IIT'vII2.
(3) Suppose that (u : Tv) = 0 for all v E V. Then (T'u : v) = 0 for all v E V. Thus T'u = 0. By (2) this is true if and only if Tu = 0, i.e., u E (Im T)1 if and only if u E Ker T. (4) If T2v = 0 then Tv E Ker T n Im T = (0) by (3). (5) By (1), the linear transformation T  AI is normal. Then by (2), II(T  AI)vII = 0 if and only if II(T'  AI)vII = 0, i.e., v is an eigenvector of T with eigenvalue A if and only if v is an eigenvector of T' with eigenvalue
1
0
(6.20) Theorem. Suppose that V is a finitedimensional inner product space
and T : V . V is a normal linear transformation. (1)
The minimal polynomial mT(X) is a product of distinct irreducible
factors. (2) Eigenspaces of distinct eigenvalues of T are orthogonal. (3) If V is a complex inner product space, then T is diagonalizable.
Proof. (1) Let p(X) be an irreducible factor of mT(X). We need to show
that p2(X) does not divide mT(X). Suppose it did and let v E V with p2(T)(v) = 0 but p(T)(v) # 0 (such a v E V exists by Theorem 3.7.1 and Lemma 3.7.17, or see Exercise 43 in Chapter 3). By Lemma 6.19 (1), U = p(T) is normal, and U2(v) = 0 but U(v) # 0, contradicting Lemma 6.19 (4).
(2) Suppose that Tv1 = Alvl and Tv2 = A2v2 where v1 and v2 are nonzero and Al 96 A2. Then (v1
: V2) = ((.'1  A2)1(T  A21)Vi : v2) = (Al  A2)1(vl:(T  A2)*v2) = 0,
since v2 is a eigenvector of T' with eigenvalue A2 by Lemma 6.19 (5).
4.6 Inner Product Spaces and Normal Linear Transformations
277
(3) Every irreducible polynomial over C is linear, so by (1), MT(X) is 0 a product of distinct linear factors and so T is diagonalizable.
(6.21) Corollary. (Spectral theorem) If V is a finitedimensional complex V is a normal transformation, then V inner product space and T : V has an orthonormal basis of eigenvectors of T. Proof. By Theorem 6.20 (3), T is diagonalizable and by (2) the eigenspaces are orthogonal. It is only necessary to choose an orthonormal basis of each 0 eigenspace.
(6.22) Remark. If V is a real vector space and T is normal, T may not be diagonalizable, but from Theorem 6.20 it follows that the real Jordan canonical form of T (cf. Theorem 4.57) will consist of 1by1 ordinary Jor
dan blocks or 2by2 real Jordan blocks. For example, the second case occurs
The case of selfadjoint linear transformations (which are automatically normal) is of particular importance; such transformations are diagonalizable even in the real case.
(6.23) Theorem. (Spectral theorem, selfadjoint case) Suppose that V is an inner product space and T : V + V is a self adjoint linear transformation. (1) All of the eigenvalues of T are real. (2) V has an orthonormal basis of eigenvectors of T.
Proof. We consider first the case F = C. In this case (1) is immediate from Lemma 6.19 (5) and then (2) follows as in Corollary 6.21. Now consider the case F = R. In this case, (1) is true by definition. To prove part (2), we imbed V in a complex inner product space and apply part (1). Let Vc = V ® V and make VC into a complex vector space by defining the scalar multiplication (6.1)
(a + bi)(u, v) = (au  by, bu + av).
That is, V ®0 is the real part of VC and 0 ®V is the imaginary part. Define an inner product on Vc by (6.2) ((u1, vl) : (u2, v2)) = (u1 : u2) + (v1 : v2) + i((vi : u2)  (u1 : v2))
We leave it for the reader to check that Equations (6.1) and (6.2) make Vc into a complex inner product space. Now, extend the linear transformation T to a linear transformation Tc : Vc ' VC by
Tc(u, v) = (T(u), T(v)). (In the language to be introduced in Section 7.2, VC = C ®R V and TC = lc OR T.) It is easy to check that TC is a complex linear transformation of
278
Chapter 4. Linear Algebra
VC, and in fact, Tc is selfadjoint. By part (1) applied to Tc we see that all the eigenvalues of Tc are real and mTQ(X) is a product of distinct (real) linear factors. Thus, mTa (X) E R(X ). If f (X) E R[X], then (6.3)
f(Tc)(u, v) = (f(T)(u), f(T)(v))
Equation (6.3) shows that MT(X) = mTo (X) and we conclude that T is diagonalizable. Part (2) is completed exactly as in Corollary 6.21. 0
4.7 Exercises 1.
2. 3. 4.
Suppose R is a finite ring with JR[ = s. Then show that Mm,n(R) is finite with JM,","(R)l = s". In particular, [M"(R)I = s"'. Prove Lemma I.I. Prove Lemma 1.2.
(a) Suppose that A = [al
AB=j:"1a;row{(B).
a", ] and B E M,..,, (R). Then show that
(b) Suppose
and B E Mm,n(R). Then show that BC =E", ci col,(B). This exercise shows that left multiplication of B by a row matrix pro
duces a linear combination of the rows of B and right multiplication of B by a column matrix produces a linear combination of the columns of B. 5.
Let S C M2(R) be defined by
S={[a aa,bER}. Verify that S is a subring of M2(R) and show that S is isomorphic to the 6.
field of complex numbers C. Let R be a commutative ring.
(a) If 1 < j < n prove that Ejj = P, 1 Ell P1t. Thus, the matrices E;; are all similar. (b) If A, B E M"(R) define [A, B] = AB  BA. The matrix [A, B] is called the commutator of A and B and we will say that a matrix C E Mn(R) is a commutator if C = [A, B] for some A, B E M,,(R). If i 54 j show
that E;, and E11  E are commutators. (c) If C is a commutator, show that 7r(C) = 0. Conclude that In is not a commutator in any Mn(R) for which n is not a multiple of the characteristic of R. What about 12 E M2(Z2)?
7.
If S is a ring and a E S then the centralizer of a, denoted C(a), is the set
C(a) = {b E S : ab = ba}. That is, it is the subset of S consisting of elements which commute with a.
a Verify that C(a) is a subring of S. b What is C(1).
4.7 Exercises
279
(c) Let R be a commutative ring and let S = MM(R). If A = D;(#) for 8.
,0 96 1 E R' then compute C(A). A matrix N E MM(R) is nilpotent if Nk = 0 for some k.
(a) If F is field of characteristic 0 and N E MnF) is nilpotent, show that
there is a matrix A E MM(F) such that A' = I,, + N for any natu
ral number m. (Hint: The binomial series may be helpful.) Are there problems if we do not assume that char(F) = 0? (b) Let N = [o o] E M2(Q). Show that there is no matrix A E M2 (Q) such that A2 = N. 9. Show that there are infinitely many A E M2(R) such that A2 = 0. is idempotent if P2 = P. Give an example, other than 10. A matrix P E 0 or It,, of a diagonal idempotent matrix. Give an example of a nondiagonal 2 x 2 idempotent matrix. Show that if P is idempotent, then T1 PT is also idempotent for all T E GL(n, R). 11. Let A E Mn(R). We say that A has constant row sums if the sum of the entries in each row is a constant a E R, i.e En_ a, = a for 1 < i < n. We define constant column sums similarly. (a) Show that A has constant row sums if and only if
for a E R and that A has constant column sums if and only if [1 1) for 0E R. 1]A =p[1
12. 13. 14.
(b) Prove that if A and B E Mn (R) both have constant row sums, then so does AB. (c) Prove that if A and B E Mn(R) both have constant column sums, then so does AB. Prove Proposition 1.13.
Prove Lemma 1.15. Let R be a commutative ring. Verify the following formulas for the kronecker product of matrices:
(a) A®(B+C)=A®B+A®C.
(b) (A ® B)` = At ® Bt. Prove Lemma 1.20. Give an example of a function D : Mn(Z2) # Z2 which is nlinear and satisfies D(P;,A) = D(A) = D(A), but which is not alternating. 17. A matrix A E Mn(R) is symmetric if At = A and it is skewsymmetric if At = A and all the diagonal entries of A are zero. 15. 16.
(a) Let V1 be the set of symmetric matrices and V2 the set of skewsymmetric
matrices. Show that V1 and V2 are both submodules of V = MM(R). If 2 is a unit in R show that V = V1 ®V2. (b) Let A E M,,(R) be skewsymmetric. If n is odd, show that det(A) = 0. If n is even, show det(A) is a square in R. 18. 19.
20.
If P,,, is a permutation matrix, show that P,; l = P. Let R and S be commutative rings and let f : R  S be a ring homomorA phism E Mn (S). Show that f
f . If Let R be a commutative ring and let H be a subgroup of the group of units
R' of R. Let N = A E GL(n, R) : det A E H}. Prove that N is a normal subgroup of GL(n, R) and that GL(n, R)/N 95 R'/H.
280
21.
Chapter 4. Linear Algebra
(a) Suppose that A has the block decomposition A = [A. A, . Prove that det A = (det Al) (det A2 ).
(b) More generally, suppose that A = [Aij] is a block upper triangular (respectively, lower block triangular) matrix, i.e., Aii is square and Aij = 0 if i > j (respectively, i < j). Show that detA = fli(detAii). 22. If R is a commutative ring, then a derivation on R is a function b : R > R such that 5(a + b) = b(a) + b(b) and b(ab) = ab(b) + b(a)b. (a) Prove that b(al ... an) = >2 1(a, ... aiib(ai)ai+1 ... an).
(b) If b is a derivation on R and A E Mn(R) let Ai be the n x n matrix obtained from A by applying b to the elements of the ith row. Show that
b(det A) = I:n 1 det Ai. 23.
If A E Mn(Q[X]) then detA E Q[X]. Use this observation and your knowledge of polynomials to calculate det A for each of the following matrices, without doing any calculation. 1
(a) A=
1
1
2X2
2
2 1
3 3 1
1 1X (b) A= 24.
25. 26. 27.
28.
29.
2
3
2
3 5
1
9X2
1 1 1
1
1
2X
1
1
1
..
1
..
1
1
... mX
Let F be a field and consider the "generic" matrix [Xij] with entries in the polynomial ring F[Xij] in the n2 indeterminants Xs (1 < i, i < n). Show that det[Xij] is an irreducible polynomial in F[Xijj. (Hint: Use Laplace's expansion to argue by induction on n.) Let A E Mn(Z) be the matrix with entii(A) = 2 (1 < i < n), entij(A) = 1 if Ii  jI = 1, and entij(A) = 0 if ji  jI > 1. Compute det(A). Let An E Mn(Z) be the matrix with entii(An) = i for 1 < i < n and aij = 1 if i # j. Show that det(An) = (n  1)!. Let A E Mn(Z) be a matrix such that entij(A) = ±1 for all i and j. Show that 2n1 divides det(A). Let R be a commutative ring and let a, b E R. Define a matrix A(a, b) E Mn(R by entii(A(a, b)) = a for all i and entij(A(a, b)) = b if i j. Compute det(A a, b)). (Hint: First find the Jordan canonical form of A(a, b).) Let V(xi, ...,xn) be the Vandermonde determinant:
V(x1, ... ,xn) =
1
x1
X1
1
x X2
X2z
x1 .
.
xn1
.
2 .
1
xn
x2n
xri1 n J
(a) Prove that
detV(x1, ... xn) =
11 (xi xj). 1 n/2, then det(A) = 0. 35.
37. 38.
Prove Lemma 2.24.
Let R be any subring of the complex numbers C and let A E Mm,n(R). Then show that the matrix equation AX = 0 has a nontrivial solution X E Mn,1(R) if and only if it has a nontrivial solution X E Mn,1(C).
39. Prove Corollary 2.29. 40. Let R be a commutative ring, let A E Mm,n(R), and let B E M,,,p(R). Prove
that
Mrank(AB) < min{Mrank(A), Mrank(B)}.
41. Verify the claims made in Remark 2.30. 42. Let F be a field with a derivation D, i.e., D : F + F satisfies D(a+b) = a+b
and D(ab) = aD(b)+D(a)b. Let K = Ker(D), i.e., K = {a E F : D(a) = 0}. K is called the field of constants of D. (a) Show that K is a field.
282
Chapter 4. Linear Algebra
(b) If ui, ..., un E F, define the Wronskian of u1, ..., un E F by UI
W(ui,.. ,un)=det
D(ul)
un ...
D(un)
pni(ul) ...
43.
44.
where D' = Do o D (i times). Show that ul, ..., un E F are linearly dependent over K if and only if W (ul, ... , u,,) = 0. E Mn(R) be a matrix such Let R be a commutative ring and let A = that all is not a zero divisor. If n > 2 prove that ai i 4 det(A) = det(B) where B E Mn_1(R is the matrix with ent,, = A[(1, i + 1) ] (1, j + 1)( for 1 < i, j < n  1. (This formula is sometimes called Choi's pivotal condensation formula. It provides another inductive procedure for the computation of determinants.) Prove the following facts about the adjoins of a matrix A E Mn(R) (in part
c,) assume that R = C): a If A is diagonal, then Adj(A) is diagonal.
b If A is symmetric, then Adj(A) is symmetric. c) If A is Hermitian, then Adj(A) is Hermitian. d) If A is skewsymmetric, then Adj(A) is symmetric or skewsymmetric according as n is odd or even. 45. Let R be an arbitrary ring. (a) If A E M,n,n(R) and B E Mn,y(R), show that
(op(AB))t =op(Bt)op(A`). (b) If A E Mn(R) is invertible, show that
op(A1)` = (op(A`))' 46.
Let P3=If (X)EZ[X]:degf(X) r (A) + rank(B)  n if B E Mn,m(F). 53.
Let M be a free Rmodule of finite rank, let f E EndR(M), and let g E
EndR(M) be invertible. If v E M is an eigenvector of f with eigenvalue A, show that g(v) is an eigenvector of gfg1 with eigenvalue A. 54. Let M be a finite rank free Rmodule over a PID R and let f E EndR(M). Suppose that f is diagonalizable and that M = N1 ® N2 where N1 and N2 are finvariant submodules of M. Show that g = f IN. is diagonalizable for
i=1,2.
55.
Let A E Mn(R), and suppose
CA(X)=det(XInA)=Xn+a1Xn1+ (a) Show that a1 =  Tr(A) and an = (1)n det(A). b More generally, show that a,
detA[a I al.
1)r QEQ,.n
56.
Prove the CayleyHamilton theorem for matrices A E M8(R) where R is
any commutative ring. (Hint: Apply the noncommutative division algorithm (Exercise 44 of Chapter 2) and the adjoint formula (Theorem 2.17).) 57. Let A E Mn(F) where F is a field.
284
Chapter 4. Linear Algebra
(a) Show that det(Adj(A)) _ (det(A))'' if n > 2. (b) Show that Adj(Adj(A)) _ (det(A))n2A if n > 2 and that Adj(Adj(A)) = A 58.
if n = 2.
Let F be a field and let A E Mn(F). Show that A and At have the same minimal polynomial.
59. Let K be a field and let F be a subfield. Let A E Mn(F).
Show that the minimal polynomial of A is the same whether A is considered
in M. (F) or Mn(K).
60. An algebraic integer is a complex number which is a root of a monic polynomial with integer coefficients. Show that every algebraic integer is an eigenvalue of a matrix A E Mn(Z) for some n. 61. Let A E Mn(R) be an invertible matrix.
(a) Show that det(X'In  A') = (X)ndet(A')cA(X).
b If
CA(X) = Xn +a, Xn1 + ... + anIX + an and
CA1(X) =Xn+b,X''+ +bn_,X+bn,
then show that bi = (1)'
ao=1. 62.
for 1 < i < n where we set
If A E MM(F) (where F is a field) is nilpotent, i.e., Ak = 0 for some k, prove that A' = 0. Is the same result true if F is a general commutative ring rather than a field?
63.
Let F be an infinite field and let Y = {Aj}jEJ C Mn(F) be a commuting family of diagonalizable matrices. Show that there is a matrix B E Mn(F) and a family fj(X)}jEJ C F[X] of polynomials of degree < n1 such that Aj = fj(B). (Hint: By Theorem 3.36 there is a matrix P E GL(n, F) such that P'AjP = diag(Aij, ... , Ani). Let t,, ... , to be n distinct points in F, and let
B = Pdiag(tl, ... ,tn)P'. Use Lagrange interpolation to get a polynomial fj (X) of degree _< n 1 such 64.
that fj(ti) = Aij for all i, j. Show that { fj(X)}jEJ works.) Let F be a field and V a finitedimensional vector space over F. Let S E EndF(V) and define Ads : EndF(V) EndF(V) by Ads(T) = [S, T] = ST  TS.
(a) If S is nilpotent, show that Ads is nilpotent. b If S is diagonalizable, show that Ads is diagonalizable. 65. Let Ni, N2 E M,(F) be nilpotent matrices. Show that N, and N2 are similar if and only if rank(Ni) = rank(N2) for all k > 1. 66. Let F be an algebraically closed field and V a finitedimensional vector space
over F.
(a) Suppose that T, S E EndF(V). Prove that T and S are similar if and only if
dim(Ker(T  A1V )k) = dim(Ker(S  Alv)k) for all A E F and k E N. (This result is known as Weyr's theorem.)
4.7 Exercises
285
If T (b) Suppose that T E Endp (V) has Jordan canonical form ® 1 is invertible, show that T1 has Jordan canonical form ®i 1JaT1 ,,, 67.
Let F be an algebraically closed field and V a finitedimensional vector space over F. (a) If T E EndF(V), show that there is a basis B = {vl, ... ,v.,} of V such
that Vi = (vi, ... , vi) is a Tinvariant subspace of V. Conclude that
T)a is an upper triangular matrix. (Hint: Since F is algebraically closed has an eigenvalue, say Al with associated eigenvector v1. Choose a complementary subspace W to (v1), define Ti E EndF (W) by T1 = ,r o T where it : V . W is the projection. Now argue by induction.) (b) If T, S E EndF(V) commute then there is a basis B of V such that both [T]1s and [S)e are upper triangular. (c) Show that the converse of part (b) is false by finding two upper triangular matrices which do not commute. (d) While the converse of part (b) is not true, show that two upper triangular matrices are "almost commutative in the following sense. Verify that
the commutator matrix [A, B] = AB  BA of two upper triangular
matrices is a nilpotent matrix. 68. Find matrices in M3(Q) with minimal polynomials X, X2, and X3. 69. Find the characteristic and minimal polynomial of each of the following matrices:
0 0 a (a)
1
1
70.
0
b
01
(b)
0
1
1
0 0
0
0 0
(c)
1
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
(F a field) has characteristic polynomial
If A E
CA(X) = X2(X  1)2(X2  1), 71.
what are the possibilities for the minimal polynomial mA(X)? Let V be a vector space and T : V + V a linear transformation. Assume that MT(X) is a product of linear factors. Show that T can be written as a sum T = D + N where D is a diagonalizable linear transformation, N is a nilpotent linear transformation, and DN = ND. Note that the hypotheses are always satisfied for an algebraically closed field (e.g., C).
72. Show that the matrices 0
1
00 1
73.
0
0 1
and
r1 I0
1
0
1
1
0
0
1]
0
are similar in M3 Z3). For each of the following matrices with entries in Q, find III the characteristic polynomial; 2 the eigenvalues, their algebraic and geometric multiplicities; 3 bases for the eigenspaces and generalized eigenspaces; 4 Jordan canonical form (if it exists) and basis for V = Q° with respect to which the associated linear transformation has this form;
(5) rational canonical form and minimal generating set for VT as Q[X)module. 0
(a) 3
(c)
0 0
4]
4
2 4 1 2
4
1 2
(b) (d)
[4
91
6
1
2
2 2
1
2 2
2
1
]
Chapte r 4. Lin ear Algebra
286
(e)
2
0
1
1
0 0
2
0 0
1
2
0
0
1
1
1
1
4
0 3
2
0
(f)
2
1
1
0
5
0
2 0 1
'
3
74.
In each case below, you are given some of the following information for a linear transformation T : V . V, V a vector space over the complex numbers C: (1) characteristic polynomial for T; (2) minimal polynomial for T; (3) algebraic multiplicity of each eigenvalue; (4) geometric multiplicity of each eigenvalue; (5) rank(VT) as an C[X]module; (6) the elementary divisors of the module VT. Find all possibilities for T consistent with the given data (up to similarity) and for each possibility give the rational and Jordan canonical forms and the rest of the data. (a) CT(X) = (X  2)4(X  3)2. (b) CT(X) = X2(X  4)' and MT(X) = X(X  4)3. (c) dim V = 6 and MT(X) = (X + 3)2(X + 1)2. d) CT X = X(X  1)4(X  2)5 , v m(1) = 2, and v`eom(2) = 2. e cT(X = (X  5)(X  7)(X (X  11).
75.
Recall that a matrix A E MM(F) is idempotent if A2 = A. (a) What are the possible minimal polynomials of an idempotent A?
f)
76.
dimV=4andmr(X)=X1.
b If A is idempotent and rank A = r, show that A is similar to B = Ir ®Onr If T : Cn Cn denotes a linear transformation, find all possible Jordan
canonical forms of a T satisfying the given data: (a) cT(X) = (X  4)3(X  5)2.
(b) n=6andmT(X)=(X 9)3.
(c) n = 5 and mr(X) _ (X  6)2(X  7).
77.
78. 79.
80. 81. 82.
83.
d T has an eigenvalue 9 with algebraic multiplicity 6 and geometric multiplicity 3 (and no other eigenvalues). (e) T has an eigenvalue 6 with algebraic multiplicity 3 and geometric multiplicity 3, and eigenvalue 7 with algebraic multiplicity 3 and geometric multiplicity 1 (and no other eigenvalues). (a) Show that the matrix A E M3(F) (F a field) is uniquely determined up to similarity by CA(X) and mA(X). (b) Give an example of two matrices A, B E M4(F) with the same characteristic and minimal polynomials, but with A and B not similar. Let A E If all the roots of the characteristic polynomial CA(X) are real numbers, show that A is similar to a matrix B E MM(R). Let F be an algebraically closed field, and let A E Mn(F). (a) Show that A is nilpotent if and only if all the eigenvalues of A are zero. are the eigenvalues b Show that Tr(Ar) = ai + + an where of A counted with multiplicity. (c) If char(F) = 0 show that A is nilpotent if and only if Tr(A') = 0 for all r E N. (Hint: Use Newton's identities, Exercise 61 of Chapter 2.) Prove that every normal complex matrix has a normal square root, i.e., if A E M. (C) is normal, then there is a normal B E A. (C) such that B2 = A. Prove that every Hermitian matrix with nonnegative eigenvalues has a Hermitian square root. Show that the following are equivalent: a) U E M,,(C) is unitary. b) The columns of U are orthonormal. c) The rows of U are orthonormal.
(a) Show that every matrix A E M,(C) is unitarily similar to an upper triangular matrix T, i.e., UAU' = T, where U is unitary. (b) Show that a normal complex matrix is unitarily similar to a diagonal matrix.
4.7 Exercises
287
Show that a commuting family of normal matrices has a common basis of orthogonal eigenvectors, i.e., there is a unitary U such that UA3U` = D3 for all Aj in the commuting family. (D3 denotes a diagonal matrix.) 85. A complex matrix A E M,,(C) is normal if and only if there is a polynomial 84.
f (X) E C[X] of degree at most n  1 such that A' = f (A). (Hint: Apply
Lagrange interpolation.) Show that B = ®A; is normal if and only if each Al is normal. Prove that a normal complex matrix is Hermitian if and only if all its eigenvalues are real. 88. Prove that a normal complex matrix is unitary if and only if all its eigenvalues have absolute value 1. 86. 87.
89.
Prove that a normal complex matrix is skewHermitian (A' = A) if and
only if all its eigenvalues are purely imaginary. 90. If A E let H(A) = I (A + A') and let S(A) = (A  A'). H(A) is z called the Hermitian part of A and S(A) is called the skewHermitian part
of A. These should be thought of as analogous to the real and imaginary parts of a complex number. Show that A is normal if and only if H(A) and
S(A) commute. Let A and B be selfadjoint linear transformations. Then AB is selfadjoint if and only if A and B commute. 92. Give an example of an inner product space V and a linear transformation T : V + V with T'T = lv, but T not invertible. (Of course, V will necessarily be infinite dimensional.) 93. (a) If S is a skewHermitian matrix, show that I  S is nonsingular and the matrix 91.
U=(I+S)(IS)1
is unitary. (b) Every unitary matrix U which does not have 1 as an eigenvalue can be written as
U = (I + S)(I  S)1 94.
for some skewHermitian matrix S. This exercise will develop the spectral theorem from the point of view of projections.
(a) Let V be a vector space. A linear transformation E : V * V is called a projection if E2 = E. Show that there is a onetoone correspondence between projections and ordered pairs of subspaces (V1, V2) of V with V1ED V2=V given by
E H (Ker(E), Im(E)). (b) If V is an inner product space, a projection E is called orthogonal if E _
E. Show that if E is an orthogonal projection, then Im(E)' = Ker(E) and Ker(E)1 = Im(E), and conversely. (c) A set of (orthogonal) projections {E1, ... , E,.} is called complete if
E;Ej=O ori$jand
lv Show that any set of (orthogonal) projections {E1, ... , E,.} with EiEE _ 0 for i 54 j is a subset of a complete set of (orthogonal) projections. (d) Prove the following result. Let T : V > T be a diagonalizable (resp., normal) linear transformation on the finitedimensional vector space V. Then there is a unique set of
288
Chapter 4. Linear Algebra
distinct scalars {Al, ... , A,.} and a unique complete set of projections (reap., orthogonal projections) {E,, ... , Er} with
T=AiEi+...+ArEr. are the eigenvalues of T and {1m(E;)};al are Also, show that the associated eigenapaces.
(e) Let T and {E;) be as in part (d) Let U : V  ' V be an arbitrary
linear transformation. Show that U = UT if and only if E1U = UEt for 1 < i < r. (f) Let F be an infinite field. Show that there are polynomials pi(X) E FIX) for 1 < i < r with pi (T) = E. (Hint: See Exercise 63.)
Chapter 5
Matrices over PIDs
5.1 Equivalence and Similarity Recall that if T; : V V (i = 1, 2) are linear transformations on a finitedimensional vector space V over a field F, then T, and T2 are similar if and only if the F[XJmodules VT, and VT, are isomorphic (Theorem 4.4.2). Since
the structure theorem for finitely generated torsion F[X]modules gives a criterion for isomorphism in terms of the invariant factors (or elementary divisors), one has a powerful tool for studying linear transformations, up to similarity. Unfortunately, in general it is difficult to obtain the invariant factors or elementary divisors of a given linear transformation. We will approach the problem of computation of invariant factors in this chapter by studying a specific presentation of the F[X]module VT. This presentation will be used to transform the search for invariant factors into performing elementary row and column operations on a matrix with polynomial entries. We begin with the following definition. (1.1) Definition. If R is a ring and M is a finitely generated Rmodule, then a finite free presentation of M is an exact sequence (1.1)
R"'
0
R^
M
0.
Note that M Coker(O) and the free presentation is essentially an explicit way to write M as a quotient of a finiterank free Rmodule by a finiterank submodule. While every finitely generated Rmodule has a free presentation (Definition 3.4.15), it need not be true that every finitely generated Rmodule has a finite free presentation as in Definition 1.1; however, if the ring R is a PID, then this is true. (1.2) Lemma. Let R be a PID and let M be a finitely generated Rmodule with p(M) = n. Then there is a finite free presentation (1.2)
withmhj(X)Xe, = E hj(X)aijei
j=1
i,j=1
and since {ei, ... , en} is a basis of F[X]n, it follows that n
hi(X)'X =Ehj(X)a;j.
(1.8)
j=1
If some hi (X) 96 0, choose i so that hi (X) has maximal degree, say,
deghi(X) = r > deghj(X)
(1 < j < n).
Then the lefthand side of Equation (1.8) has degree r + 1 while the righthand side has degree < r. Thus hi(X) = 0 for 1 < i < n and C is a basis of K. We can summarize the above discussion as follows:
(1.5) Proposition. Let V be a vector space of dimension n over the field F, let T : V * V be a linear transformation, and let 8 be a basis of V. Then there is a finite free presentation of VT (1.9)
0 
' F[X]n ±24 F[X]n + VT .0
in which the matrix of 08 in the standard basis of F[X]n is
XIn  [T]8. Proof. If O8(ej) = pj(X) as in Equation (1.5), then pj(X) = colj(XIn  A) where A = [T]8. Thus [O8]A = XIn  A, and sequence (1.9) is exact by Lemma 1.4.
According to our analysis, VT = Coker(O8). It is worthwhile to see this isomorphism in a simple explicit example.
5.1 Equivalence and Similarity
293
(1.6) Example. Let F be a field, let V = F2, and let T : V + V be defined
by T(ul, u2) = (u2i 0) (cf. Example 3.1.5 (13)). If B = (vi, v2) is the standard basis on F2, then '+,e(f (X), 9(X)) = f (T)(vi) + g(T)(v2) = aov1 + bov2 + blvl where
f(X)=ao+a1X+ g(X) = bo+b1X + +b,,,Xm.
Since A=[T]B= [oo],wehave X12A= [° X) so that 013 L(1f(x)1) 9(X) l
Note that
(f(X)1 l g(X) J
 [xf(x)_9(X)1 X9(X)
 lao+bll + (Xf(X)9'(X) Xg(X) bo
L
JJ
where
9'(X) =
9(X)  bo
X f(X) +9(X)  ao  bl f(X) = X Since f (X) and g(X) are arbitrary, we see that F[X]2/Im(OB) F2 (1.10) as Fmodules, while as an F[X]module, r
F[X]21 Im(s8) L F[X]
11)
J
101.
Equation (1.11) follows from Equation (1.10), the observation
X [0] 1
and the fact that
0J+I
X1,
E Im(OB). It follows immediately that
I
XJ L
F[X]2/ Im(OB) ?` VT
as F[X]modules.
(1.7) Theorem. Let F be a field and let A, B E
be matrices. Then
A is similar to B if and only if the polynomial matrices XI,  A and XI,,  B E M (F[X ]) are equivalent.
294
Chapter 5. Matrices over PIDs
Proof. If A and B are similar, then B = P'AP for some P E GL(n, F). Then
(XIn  A) (XIn  B) are equivalent in Mn(F(X]). Conversely, suppose that (XIn  A) and (XIn  B) are equivalent in Mn(F[X]). Thus there are matrices P(X), Q(X) E GL(n, F[X]) such that (1.12)
(XI,  B) = P(X)(XIn  A)Q(X).
Now, if V is a vector space of dimension n over F and B is a basis of V, then there are linear transformations T1 and T2 on V such that [T1]B = A and [T2]B = B. By Proposition 1.5, there are injective homomorphisms
Oi : F[X]n ~ F[X]n such that Coker(0i)
VT, and the matrix of ¢i with respect to the standard
basis on F[X]n is (XIn  A) and (XI,,  B) for i = 1, 2 respectively. By Equation (1.12), the F[X]module homomorphisms 01 and 02 are equivalent. Therefore, Proposition 1.3 applies to give an isomorphism of VT, and VT, as F[X]modules. By Theorem 4.4.2, the linear transformations T1 and T2 are similar, and taking matrices with respect to the basis B shows that A and B are also similar, and the theorem is proved. This theorem suggests that a careful analysis of the concept of equivalence of matrices with entries in F[X] is in order. Since it is no more difficult to study equivalence of matrices over any PID R, the next two sections will
be devoted to such a study, after which, the results will be applied to the computation of invariant factors and canonical forms for linear transformations. We will conclude this section by carefully studying the relationship between generating sets of a module M that has two different finite free presentations.
(1.8) Example. Let R be a commutative ring and let M be an Rmodule with two equivalent finite free presentations. That is, there is a commutative diagram of Rmodules and homomorphisms Rm
(1.13)
jai
Rm g
M '
0
11M
M * 0
where h1 and h2 are isomorphisms. If A = {el, ... en} is the standard basis on Rn, then V = {vl, ... , vn} and W = {w1, ... , wn} are generating sets of the Rmodule M, where vi = 9r1(ei) and w3 = ir2(e,). Note that we are not assuming that these generating sets are minimal, i.e., we do not assume that µ(M) = n. From the diagram (1.13) we see that the generators v; and w3 are related by
5.1 Equivalence and Similarity (1.14)
295
wj = 7r2(ej) = ir1(h2 1(ej))
Let us analyze this situation in terms of matrix representations of the homomorphisms. All matrices are computed with respect to the standard bases
on R' and R. Thus, if A = [f], B = [g], Q = [hi], and P = [h2], then the commutativity of diagram (1.13) shows that B = PAQ1. Furthermore, if P1 = [pt], then Equation (1.14) becomes wj = ir1(h21(ej)) n
_ 11pijei) i=1
=
(1.15)
tPiv.
That is, wj is a linear combination of the vi where the scalars come from the jeh column of P1 in the equivalence equation B = PAQ1. (1.9) Example. We will apply the general analysis of Example 1.8 to a specific numerical example. Let M be an abelian group with three generators v1i v2, and v3, subject to the relations 6v1 + 4v2 + 2v3 = 0 2v1 + 2v2 + 6v3 = 0.
That is, M = Z3/K where K is the subgroup of Z3 generated by yi = (6,4,2) and y2 = (2,2,6), so there is a finite free presentation of M
0, Z2
TA
(1.16)
+ Z3
11
+ M>0
where TA denotes multiplication by the matrix 6
A= 4
2
2 2 6
We wish to find B = PAQ equivalent to A where B is as in Proposition 4.3.20. If
P=
0
0
1
0
1
1
2
2
and
1
_
1
Q=
[0
3
1
'
then
2
B=PAQ= 0
0 10
0
0
(We will learn in Section 3 how to compute P and Q.) Then
296
Chapter 5. Matrices over PIDs
P1=
3 2
2
1
1
0
1
0
0
Therefore, w1 = 3v1 + 2v1 + v3, w2 = 2v1 + V2, and w3 = v3 are new generators of M, and the structure of the matrix B shows that 2w1 = 0 (since 2w1 = 6v1 + 4v2 + 2v3 = 0), lOw2 = 0, and w3 generates an infinite cyclic subgroup of M, i.e.,
MZ2®Z1o®Z.
5.2 Hermite Normal Form In this section and the next, we will be concerned with determining the simplest form to which an m x n matrix with entries in a PID can be reduced using multiplication by unimodular matrices. Recall that a unimodular matrix is just an invertible matrix. Thus, to say that A E M,, (R) is unimodular is the same as saying that A E GL(n, R). Left and right multiplications by unimodular matrices gives rise to some basic equivalence relations, which we now define.
(2.1) Definition. Let R be a commutative ring and let A, B E M"',n(R). (1)
We say that B is left equivalent to A, denoted B £ A, if there is a
unimodular matrix U E GL(m, R) such that B = UA. R (2) We say that B is right equivalent to A, denoted B A, if there is a unimodular matrix V E GL(n, R) such that B = AV. (3) We say that B is equivalent to A, denoted B N A, if there are unimodular matrices U E GL(m, R) and V E GL(n, R) such that B = UAV. Each of these relations is an equivalence relation, and we would like to compute a simple form for a representative of each equivalence class in an algorithmic manner. (Equivalence of matrices has been introduced in Definition 4.3.18, and, in fact, if R is a PID then we have described the equivalence classes of matrices under the relation of equivalence in Proposition 4.3.20. What we will concentrate on at the present time are algorithmic aspects of arriving at these representatives.) This will require the ability to
construct a number of unimodular matrices. The elementary matrices introduced in Definition 4.1.8 provide some examples of unimodular matrices over any commutative ring, and we will see that for a Euclidean domain, every unimodular, i.e., invertible, matrix is a product of elementary matrices. The following fundamental result describes an inductive procedure for constructing unimodular matrices with a prescribed row or column.
5.2 Hermite Normal Form
297
(2.2) Theorem. Let R be a PID, let a1, ..., a,a E R, and let d = gcd{a1, ... ,an}. Then there is a matrix A E Mn(R) such that (1) rowl(A)=[a1
an], and
(2) det(A) = d.
Prof. The proof is by induction on n. If n = 1 the theorem is trivially true. Suppose the theorem is true for n  1 and let Al E Mn1(R) be a an_1] and det(A1) = d1 = gcd{a1,..., matrix with rowl(A1) = [a1 an11 Since d = gcd{al,...,an}
= gcd{gcd{a1i ... a, _1}, an} = gcd{dl, an},
it follows that there are u, v E R such that ud1  van = d. Now define A by AI
an 0
A= 0 L
U
ai for 1 < i < n  1, it follows that A E M. (R) and row, (A) = an ]. Now compute the determinant of A by cofactor expansion along the last column. Thus, Since d1 [a1
det(A) = udet(A1) + (1)n+1an det(Ain) where A1n denotes the minor of A obtained by deleting row 1 and column n. an_1 ] Note that A1n is obtained from AI by moving row,(A1) = (a1 to the n 1 row, moving all other rows up by one row, and then multiplying the new row n 1 (= [ a1 an I ]) by v/d1. That is, using the language of elementary matrices
A1n = Dn1(v/d1)Pn1,n2 ... P3.2 P21A1. Thus,
det A1n = (i) (1)n2 det Al = (1)n2V. Hence,
det A = u det A 1 + (1)n+1 an det Ain (1)n+lan(1)n2v
= ud1 +
= ud1  Van = d.
Therefore, the theorem is true for all n by induction.
0
Chapter 5. Matrices over PIDs
298
(2.3) Remark. Note that the proof of Theorem 2.2 is completely algorithmic, except perhaps for finding u, v with ud,  van = d; however, if R is a Euclidean domain, that is algorithmic as well. It is worth noting that the existence part of Theorem 2.2 follows easily from Theorem 3.6.16. The details are left as an exercise; however, that argument is not at all algorithmic.
(2.4) Corollary. Suppose that R is a PID, that a1,
..., a, are
relatively
prime elements of R, and 1 < i < n. Then there is a unimodular matrix A2 with rowi(A2) _ [a, an ] and a unimodular matrix B2 with col2(B2) _ [a,
...
an]
t
.
Proof. Let A E Mn(R) be a matrix with
...
row, (A) = [a,
an ]
and det A = 1, which is guaranteed by the Theorem 2.2. Then let Ai = P12A and let B2 = Al P1. where P12 is the elementary permutation matrix that (] interchanges rows 1 and i (or columns 1 and i).
(2.5) Example. We will carry through the construction of Theorem 2.2 in a specific numerical example. Thus let R = Z and construct a unimodular matrix A E GL(3, Z) with
row,(A) _ [25
7].
15
3 z Then 3 5  2 7 = 1,
Since 2.25315 =5, we may take A, so the induction step will give 25
15
7
3
2
0
10
6
3
A=
Then det A = 1, so A is unimodular. Furthermore, if we wish to compute a unimodular matrix B E GL(4, Z) with
row,(B) = [25
15
7
9]
then we may use the matrix A in the induction step. Observe that 109 = 1, so the algorithm gives us a matrix
B
25
15
7
9
3
2
0
10
6
3
0 0
25
15
7
10
that is unimodular and has the required first row.
5.2 Hermite Normal Form
299
(2.6) Lemma. Let R be a PID, let A # 0 E M,,,,1(R), and let d = gcd(A) (i.e., d is the gcd of all the entries of A). Then there is a unimodular matrix U E GL(m, R) such that d
UA=
0
0
Moreover, if R is a Euclidean domain, then U can be taken to be a product of elementary matrices. .. am it Proof We may write bla1 + + bmam = d 0 0 where A = [ al and b1, ..., bm E R. Then
bt\d/+.+bm(d)=1 so {b1, ... bm} is relatively prime. By Theorem 2.2 there is a matrix U1 E GL(m, R) such that rows (Ul) = [ b1 ... bm ]. Then
+ uimam so that d I ci for all i > 2. Hence ci = aid for ai E R. Now, if U is defined by
and ci = uilal +
U = T21(a2)T31(a3) ... Tml(am)Ul then Proposition 4.1.12 shows that
This completes the proof in the case of a general PID. Now suppose that R is a Euclidean domain with Euclidean function v : R \ {0}  Z+. We shall present an argument, which is essentially a second proof of Lemma 2.6 in this case. This argument is more constructive in that only elementary row operations, i.e., left multiplications by elementary matrices, are used. Hence the U constructed will be a product of elementary matrices. Let v(A) = min{v(at) : 1 < j < m} and suppose that v(A) = Qm ]t has v(/31) < v(,0j) for j > 2. Each Qi v(ai). Then P1iA = 10, can be written as Qi = 7it1 + ri where ri = 0 or v(ri) < v(l31). Therefore, subtracting ryi row1(P1iA) from rowi(P1iA) gives a matrix
300
Chapter 5. Matrices over PII)s
al
Al = T2,(72) ... T,.1(7m)PjjA =
rl
rm
where ri = 0 or v(rj) < v(pl). If some rj # 0 then we have found a matrix Al left equivalent to A via a product of elementary matrices for which v(A1) < v(A). We can repeat the above process to find a sequence of matrices
amp
left equivalent to A via a product of elementary matrices such that (1) v(aiill) > v(a(')) for all i > 1, and v(a(')) for j > 2. 0 or (2)
Since condition (1) can only occur for finitely many A;, it follows that we must have a3(') = 0 for j > 2 for some A;, i.e.,
al') 1
UA=A,=
0 0
for some U E GL(m, R), which is a product of elementary matrices. It remains to observe that a(') = b is a gcd of the set {a1, ... ,am}. But the equation b0
am
0
shows that b E (a 1, ... , am) and b
[ai] = U1 am
U
in
shows that (al, ... am) C (b), i.e., (b) = (a1, proved.
, am) and the lemma is 0
Given an equivalence relation ti on a set X, a complete set of representatives of  is a subset P C X such that P has exactly one element from
5.2 Hermite Normal Form
301
each equivalence class. Thus, P is a complete set of representatives of  if each x E X is equivalent to a unique a E P. The cases we wish to consider concern some equivalence relations on a commutative ring R. A complete set of nonassociates of R is a subset P C R such that each element of R is an associate of a unique b E P, i.e., if a E R then there is a unique b E P and a unit u E R* such that b = au. Similarly, if I C R is an ideal, then a complete set of residues modulo I consists of a subset of R, which contains exactly one element from each coset a + I. If I is the principal ideal Ra for some a E R, then we speak of a complete set of residues modulo a. (2.7) Examples. (1)
(2)
(3)
If R = Z, then a complete set of nonassociates consists of the nonnegative integers; while if m E Z is a nonzero integer, then a complete set of residues modulo m consists of the m integers 0, 1, ... , Iml  1. A complete set of residues modulo 0 consists of all of Z. If F is a field, then a complete set of nonassociates consists of {0, 1}; while if a E F \ {0}, a complete set of residues modulo a is {0}. If F is a field and R = F[X] then a complete set of nonassociates of R consists of the monic polynomials together with 0.
(2.8) Definition. Let R be a commutative ring, let P C_ R be a complete set
of nonassociates of R, and for each a E R let P(a) be a complete set of residues modulo a. Then a matrix A = [aij] E M,n,n(R) is said to be in Hermite normal form (following P, P(a)) if A = 0 or A 0 and there is an integer r with 1 < r < m such that 0 for 1 < i < r, rowi(A) = 0 for r + 1 < i < m; and (1) rowi(A) < n,. < m such there is a sequence of integers 1 < nl < n2 < (2)
that aij = 0 for j < ni (1 < i < r), ain, E P\101 (1 < i < r), and ajnj E P/lain;) for 1 < j < i. Thus, if the matrix A is in Hermite normal form, then A looks like the following matrix: Table 2.1. Hermite normal form 0 0
0
...
0
al.,
*
...
*
0
...
0
aln3 a2n3
* *
...
a2n,.
...
0
* *
alnr
0
al2 a2n2
...
...
0
0
0
...
0
0
0
a3n3
*
...
a3nr
arn r 0
0 0
... ...
0 0
...
0
...
...
0
...
...
*
...
0
0
302
Chapter 5. Matrices over PIDs
where * denotes an entry that can be any element of R. If R is a field and P = {0, 11 while P(a) = {0} for every a 96 0, then in the Hermite normal form we will have a,,,, = I while ajn, = 0 if j < i. The resulting matrix is what is usually called a reduced row echelon matrix and it is used to solve systems of linear equations Ax = b over a field. Our main result on left equivalence of matrices is that if R is a PID then every matrix is left equivalent to one in Hermite normal form. (2.9) Theorem. Let R be a PID, P C R a complete set of nonassociates, and P(a) (a E R) a complete set of residues modulo a. Then any A E M,a,n(R) is left equivalent to a matrix H in Hermite normal form. If R happens to be Euclidean, then H = UA where U is a product of elementary matrices.
Proof. The proof is by induction on the number of rows m. If m = 1 and A 34 0 let n1 be the first index with al,,, 0. Then let ualn, = bin, E P. Then B = uA is in Hermite normal form. Now suppose that m > 1 and that every matrix in Mm_1,,,(R) (for arbitrary n) is left equivalent (using a product of elementary matrices if R is Euclidean) to a matrix in Hermite normal form. Let n1 be the smallest integer such that coln, (A) $ 0. Let (cola, (A))' = [ a1 . a,a) and let d = ged{al, ... , a,,,}. Then d 36 0 and, by Lemma 2.6, there is an invertible matrix U1 E GL(m, R) (which may be taken as a product of elementary matrices if R is Euclidean) such
that d
Al=U1A=
*
0
B, 0
where B1 E M,_l,n_1(R). By the induction hypothesis, there is an invertible matrix V E GL(m  1, R) (which may be taken as a product of elementary matrices if R is Euclidean) such that VB1 is in Hermite normal form. Let 1
U2
0
0 V
Then U2A1 = A2 is in Hermite normal form except that the entries a,,,,
(i > 1) may not be in P(a;a,). This can be arranged by first adding a multiple of row 2 to row 1 to arrange that a,,,, E P(a2n, ), then adding a multiple of row 3 to row 1 to arrange that a1n3 E P(a3n, ), etc. Since a;3 = 0 if j < n;, a later row operation does not change the columns before n;, so at the end of this sequence of operations A will have been reduced to Hermite normal form, and if R was Euclidean then only elementary row operations will have been used. 0 If we choose a complete set of nonassociates for R so that it contains 1 (as the unique representative for the units) and a complete set of representatives modulo 1 to be {0}, then the Hermite form of any U E GL(n, R) is
5.2 Hermite Normal Form
303
I,,. This is easy to see since a square matrix in Hermite normal form must be upper triangular and the determinant of such a matrix is the product of the diagonal elements. Thus, if a matrix in Hermite normal form is invertible, then it must have units on the diagonal, and by our choice of 1 as the representative of the units, the matrix must have all 1's on the diagonal. Since the only representative modulo 1 is 0, it follows that all entries above the diagonal must also be 0, i.e., the Hermite normal form of any U E GL(n, R) is I,,. If we apply this observation to the case of a unimodular matrix U with entries in a Euclidean domain R, it follows from Theorem 2.9 that U can be reduced to Hermite normal form, i.e., I,,, by a finite sequence of elementary row operations. That is,
El...E1U=I where each Ej is an elementary matrix. Hence, U = El 1 . . . ET' is itself a product of elementary matrices. Therefore, we have arrived at the following result. (2.10) Theorem. Let R be a Euclidean domain. Then every invertible matrix over R is a product of finitely many elementary matrices. Proof.
(2.11) Remark. If R is not Euclidean then the conclusion of Theorem 2.10 need not hold. Some explicit examples of matrices in GL(2, R) (R a PID), which cannot be written as a product of elementary matrices, have been given by P. M. Cohn in the paper On the structure of the GL2 of a ring, Institut des Hautes Etudes Scientifiques, Publication #30 (1966), pp. 554. A careful study of Lemma 2.6 shows that the crucial ingredient, which Euclidean domains have that general PIDs may not have, is the Euclidean algorithm for producing the gcd of a finite subset of elements.
(2.12) Example. Let R = Z, P = Z+, and for each m 54 0 E Z, let P(m) _ {0, 1, ... , Imp  1}.
Let P(0) = Z. Thus we have chosen a complete set of nonassociates for Z and a complete set of residues modulo in for each m E Z. We will compute a Hermite normal form for the integral matrix 2
A= 6
3
9 4
8
4
1
4
5
3
1
The left multiplications used in the reduction are U1, ..., U7 E GL(3, Z), while Al = U1A and A; = U;A;_1 for i > 1. Then
304
Chapter 5. Matrices over PIDs
1
1
0
0
1
0
0
0 0
1
UI =
1
U2= 3 4
r1
U3=
U4=
U5 =
U8=
1
0
1
0 0
0
1
0
0
10
9
1
1
1
0
0
0
1
0
0
2
1
1
0
0
0
1
0
0
0 1
1
5
0 0
1
0 0
0
1
0
2
1
0
0
1
U7= 10
Al = Ul A = A2 = U2A1 =
A3 = U3A2 =
A4 = U4A3 =
A5 = U5A4 =
1
6 8
3 4
2
1
5 2
0 0
0
19
9
0
21
7
2
1
5 2
0 0
0
1
27
0
2
2
1
5
2 2
0 0
0
1
27
2
1
0 0
0 0
2
0
4
3
1
1
56 5 2 0
27 56
1
0
1
0
0 0
1
0
133 27 56
2
1
0
21
0 0
0 0
1
27 56
A6=UBA5= 0 0
A7=U7A6=
5 2
2
0
The matrix A7 is the Hermite normal form associated to the matrix A (using the system of representatives P and P(m)). A natural question to ask is whether the Hermite normal form of A guaranteed by Theorem 2.9 is unique. Certainly one can get a different Hermite normal form by changing the complete set of nonassociates or the complete set of residues modulo a E R, but if we fix these items then the Hermite normal form is uniquely determined, independent of the precise sequence of operations needed to achieve this form. This is the content of the next result. (2.13) Theorem. Let R be a PID, P C R a complete set of nonassociates and P(a) a complete set of residues modulo a for each a E R. If A E Mm,n(R) then the Hermite normal form of A is unique.
Proof. Without loss of generality we may assume that A 0 0. First note that the number of nonzero rows in any Hermite normal matrix H, which is left equivalent to A, is just rank(A). To see this, suppose the Hermite
normal matrix H has r nonzero rows, let 1 < nl < < n,. < n be the integers guaranteed by Definition 2.8, and let a = (1, 2, ... , r) and a,.,,, 0. Thus, Q = (ni, n2, ... , n,.). Then det H[a 10) = ain, a2n2 rank(H) > r and any submatrix of H with more than r rows will have a row of zeros. Hence, rank(H) = r, and since rank is preserved by equivalence
5.2 Hermite Normal Form
305
(Proposition 4.2.36), it follows that
r = rank(H) = rank(A).
Now suppose that A L H and A L K where H and K are both in Hermite normal form. Then there is a unimodular matrix U such that H = UK, and by the above paragraph, both H and K have r = rank(A) nonzero rows. Let 1 < nl < n2 < < nr < n be the column indices for K given by the definition of Hermite normal form and let 1 < t1 < t2 < < tr < n be the column indices for H. We claim that t; = ni for 1 < i < r. Indeed, n1 is the first nonzero column of K and ti is the first nonzero column of H; but col, (H) = U col, (K) and U is invertible, so n1 = t1. Then we conclude
that for j = ni = ti, hin, 54 0, kin, 0 0, and kln,
[hi ni
0
0
U
0
0
so kin,u,i = 0 for s > 1, and hence, u,1 = 0 for s > 1. Therefore, _
U
u11
Ui
0
If H = [ro i(H)] and K = [ro i(K)] Hi KI where Hi, K1 E M,n_i,n(R) then H1 = Ui Ki and H1 and K1 are in Hermite normal form. By induction on the number of rows we can conclude that nj = tj for 2 < j < r. Moreover, by partitioning U in the block form U = [ u" u12 ] where Ul 1 E Mr (R) U2i Usz and by successively comparing cola, (H) = U Colnj (K) we conclude that U21 = 0 and that U1 i is upper triangular. Thus, U11
U12
0
U22
0
0
Uir u2r
U12
U= urr
0
and detU = u11
U22
urr(det(U22)) is a unit of R. Therefore, each uii is
a unit of R; but hin, = uiikin, for 1 < i < r so that hin, and kn, are associates. But hin, and kin, are both in the given complete system of nonassociates of R, so we conclude that hin, = kin, and hence that uii = 1 for 1 < i < r. Therefore, each diagonal element of U11 must be 1. Now
suppose that 1 < s < r  1. Then m
hs,n.+, _ E us,Yk7,n.+j f=1
= ussks,n.+i + us,a+iks+i,n.+i
306
Chapter 5. Matrices over PIN
since u,.. = 0 for y < s while k7n,+, = 0 if 'y > s + 1. Since u,, = 1, we conclude that h.,,n.+i = ks,n.+, + u,,,+1ks+1,n.+1 Therefore,
hs,n.+, = ks,n.+,
(mod k,+1,n.+, ),
and since h,,n,+, and k,.n,+, are both in P(k,+l,n,+, ), it follows that hs,n.+, = k,,n,+, and, hence, us,s+1 = 0 for I < s < r  1. We now proceed by induction. Suppose that u,,,+j = 0 for 1 < j < r1 (we just verified that this is true for j = 1) and consider
f=l
= us,s+j+1 ks+j+1,n,+j+, + u,,k,,n,+,+, us,s+j+1ks+j+1,n,+,+1 + ks,n.+,+t
Therefore, h,,,,,+,+, = k,,n,+,+, since they both belong to the same residue Hence u,,,+j+I = 0. Therefore, we have shown class modulo that U has the block form k,+j+1,n.+,r,
U _ rIr 0
U12 U22
and, since the last m  r rows of H and K are zero, it follows that H = 0 UK = K and the uniqueness of the Hermite normal form is proved. We will conclude this section with the following simple application of the Hermite normal form. Recall that if R is any ring, then the (twosided) ideals of the matrix ring M,, (R) are precisely the sets M,' (J) where J is an ideal of R (Theorem 2.2.26). In particular, if R is a division ring then the only ideals of M,, (R) are (0) and the full ring M,, (R). There are, however,
many left ideals of the ring M,,(R), and if R is a PID, then the Hermite normal form allows one to compute explicitly all the left ideals of M,(R), namely, they are all principal.
(2.14) Theorem. Let R be a PID and let J C Mn(R) be a left ideal. Then there is a matrix A E Mn(R) such that J = (A), i.e., J is a principal left ideal of Mn(R).
Proof. Mn(R) is finitely generated as an Rmodule (in fact, it is free of rankn2), and the left ideal J is an Rsubmodule of Mn(R). By Theorem 3.6.2, J is finitely generated as an Rmodule. Suppose that (as an Rmodule)
J=(B1,...,Bk) where B; E Mn(R). Consider the matrix
5.3 Smith Normal Form
307
B1
B=
E
Bk
There is an invertible matrix P E Mnk(R) such that PB is in Hermite normal form. Thus,
PB =
(2.1)
A 0
0
and if we partition P = [P,J) into blocks where P;j E Mn(R), it follows from Equation (2.1) that
A=P11B1+ 
+ P1kBk
Therefore, A E J and hence the left ideal (A) C J. Since A
B=Q
0 .
.0
where Q = P1 = [Qt
,
it follows that B; = Qi1A. Therefore, J C (A),
and the proof is complete.
0
5.3 Smith Normal Form In contrast to the relatively complicated nature of the Hermite normal form, if multiplication by nonsingular matrices is allowed on both the left
and the right, then one can reduce a matrix A over a PID R to a particularly simple form. The existence of this simple diagonal form, known as the Smith normal form, has essentially already been proved in Proposition 4.3.20. Combining this with Theorem 2.10 will provide a reasonably efficient procedure for the computation of the invariant factors of a linear transformation. Additionally, this same computational procedure produces the change of basis map that puts a matrix (or linear transformation) in rational canonical form. We will also consider applications of the Smith normal form to the solution of linear diophantine equations.
(3.1) Theorem. Let R be a PID and let A E M,n,n(R). Then there is a U E GL(m, R) and a V E GL(n, R) such that
Chapter 5. Matrices over PIDs
308
UAV =
(3.1)
Dr 0
0 0
where r = rank(A) and Dr = diag(sl, ... , s,.) with si 0 0 (1 < i < r) and Si I si+l for 1 < i < r  1. Furthermore, if R is a Euclidean domain, then the matrices U and V can be taken to be a product of elementary matrices.
Proof. Consider the homomorphism TA : R"  Rm defined by multiplication by the matrix A. By Proposition 4.3.20, there is a basis B of R' and
a basis C of R' such that (3.2)
[TA]"
_
[Dr
0]
where Dr = diag(sl, ... , Sr) with si $ 0 (1 < i < r) and si
I
si+l for
1 < i < r  1. If C' and B' denote the standard bases on R' and R" respectively, then the change of basis formula (Proposition 4.3.16) gives
]=
(3.3) [ D0r
[ZA]C = PC [TA]C, Phi
00
Since A = [TA]C, and since the change of basis matrices are invertible, Equation (3.3) implies Equation (3.1). The last statement is a consequence of Theorem 2.10.
(3.2) Remarks.
(1) The matrix
[r
00] is called the Smith normal form of A after H. J.
Smith, who studied matrices over Z. (2) Since the elements s1, ... , Sr are the invariant factors of the submodule
Im(TA), they are unique (up to multiplication by units of R). We shall
call these elements the invariant factors of the matrix A. Thus, two matrices A, B E M,,n(R) are equivalent if and only if they have the same invariant factors. This observation combined with Theorem 1.7 gives the following criterion for the similarity of matrices over fields.
(3.3) Theorem. Let F be a field and let A, B E Mn(F). Then A and B are similar if and only if the matrices XI7,  A and XI,  B E Mn(F[X]) have the same invariant factors. Proof.
(3.4) Remark. If R is a Euclidean domain then A can be transformed into Smith normal form by a finite sequence of elementary row and column operations since every unimodular matrix over R is a finite product of elementary matrices (Theorem 2.10). It is worthwhile to describe explicitly an algorithm by which the reduction to Smith normal form can be
5.3 Smith Normal Form
309
accomplished in the case of a Euclidean domain R with degree function v : R \ 0 + Z+. The algorithm is an extension (to allow both row and column operations) of the second proof of Lemma 2.6. If A E M,,,,,,(R) is a nonzero matrix, let v(A) = min{v(ain) : 1 < i < m; 1 < j < n}.
By using a sequence of row and column exchanges we can assume that v(ale) = v(A). Then if i > 1, we may write ail = alibi + bt1 where bit = 0 or v(bil) < v(a11). By subtracting birow1(A) from rowi(A) we obtain a matrix All in which every element of coli(A(')) is divisible by all or v(A(11) < v(A). If we are not in the first case, repeat the process with A replaced by AM. Since v(A) is a positive integer, this process cannot go on indefinitely, so we must eventually arrive at a matrix B with b11 bil for 2 < i < m. By applying a similar process to the elements of the first I
row, we may also assume that elementary row and column operations have produced a matrix B in which b11 bit and b11 I bl, for 2 < i < m and I
2 < j < n. Then subtracting multiples of the first row and column of B produces an equivalent matrix B = [b11J ® C where C E Mml,,,l. We may arrange that b11 divides every element of C. If thisis not the case already, then simply add a row of C to the first row of B, producing an equivalent matrix to which the previous process can be applied. Since each
repetition reduces v(B), only a finite number of repetitions are possible before we achieve a matrix B = [b11J ® C in which b11 divides every entry of C. If C is not zero, repeat the process with C. This process will end with the production, using only elementary row and column operations, of the Smith normal form.
(3.5) Example. It is worthwhile to see this algorithm in practice, so we will
do a complete example of the computation of the Smith normal form of an integral matrix. A simple method for producing U E GL(m, R) and V E GL(n, R) so that UAV is in Smith normal form is to keep track of the elementary row and column operations used in this reduction. This can be conveniently accomplished by simultaneouly applying to In each row operation done to A and to I each column operation performed on A. This process is best illustrated by a numerical example. Thus let 2
A=
1
4
1
3
1 3
4
0
1 1
I E M3 4(Z). ,
16
We shall reduce A to Smith normal form by a sequence of row and column operations, and we shall keep track of the net effect of these operations by simultaneously performing them on 13 (for the row operations) and 14 (for the column operations). We will use the arrow to indicate the passage from one operation to the next.
Chap ter 5. Matrices over PIDs
310
A
13 1
0 0
0 1
0 0
0 1
0 1
0 0 1
H 1 2 0
+
2
0
1
1
1
4
1
0
1
0
2
0
1
0
0
1
16
1
3
1
0
1 3 3
3
12
0 1
0
0
0
0
3
3
3
4
1
0
0
12
12
0
1
0
1
0
0
1
2 4
0
0
3
0 12
0
0 0 12
0
1
0
1
0
0
0
1
2
0
0
3
0
4
1
0 0
U
1
1
0
0
0
1
0
0 0
0 0
0
1
0
0
0
1
0
0 1
0 3
1
0
0 0
1
12
0
0
0 0
1
0
0
0
0 0
3
1
0
1
16
2
1
0
0
3 1
1
0
0
0
1
0
4
0
0
1
1
4 1
4
1
3 3
0
0
=
4
0
14
0
12
0
S
0
0
1
1
10 0 1
_
0
1
2
1
1
1
0
1
0
0
0 1
2
1
2
1
1
2
0
1
1
0
0
1
V
Then UAV = S and S is the Smith normal form of A. (3.6) Remark. Theorem 3.3 and the algorithm of Remark 3.4 explain the origin of the adjective rational in rational canonical form. Specifically, the
invariant factors of a linear transformation can be computed by "rational" operations, i.e., addition, subtraction, multiplication, and division of polynomials. Contrast this with the determination of the Jordan canonical form, which requires the complete factorization of polynomials. This gives an indication of why the rational canonical form is of some interest, even though the Jordan canonical form gives greater insight into the geometry of linear transformations.
The ability to compute the invariant factors of a linear transformation by reduction of the characteristic matrix XIn  [T]13 to Smith normal form has the following interesting consequence.
(3.7) Proposition. Let F be a field and let A E to the transposed matrix At.
Then A is similar
5.3 Smith Normal Form
311
Proof. Consider the matrix X In A E M.(F[XI ). Then there are invertible matrices P(X), Q(X) E GL(n, F[X]) such that
P(X) (XI.  A)Q(X) = diag(s1(X), ... , sn(X))
(3.4)
where si(X) I si+1(X) for 1 < i < n  1. Taking the transpose of Equation (3.4) shows that
Q(X)t(XIn At)P(X)t =diag(sl(X), ... sn(X))Thus, A and At have the same invariant factors and hence are similar.
This result cannot be extended to matrices over arbitrary rings, even PIDs, as the following example shows:
(3.8) Example. Let R be a PID, which is not a field, and let p E R be a prime. Consider the matrix 0
A= 0
0
p 0 0
0
E M3(R).
1
0
Claim. A is not similar to At. Proof. Suppose that T = [tip] E M3(R) satisfies the matrix equation AT = TAt. Then (3.5)
0 p
0
t11
t12
t13
0
0
1
t21
t22
t23
0
0
0
t31
t32
t33
=
t11
t12
t13
0
0
0
t21
t22
t23
p
0
0
t31
t32
t33
0
1
0
,
which implies that
(3.6)
pt21
pt22
pt23
t31
t32
t33
0
0
0
=
pt12
t13
pt22
t23
0 0
pt32
t33
0
From Equation (3.6), we conclude that t32 = t23 = t33 = 0, t12 = t21, t13 = t31 = pt22 = ps for some s E R. Therefore, T must have the form T =
tll
t12
PS
t12
s
0
Ps
0
0
and hence, det(T) = p2s3. Since this can never be a unit of the ring R, it follows that the matrix equation AT = TAt has no invertible solution T. Thus, A is not similar to At.
312
Chapter 5. Matrices over PIDs
(3.9) Remark. Theorem 1.7 is valid for any commutative ring R. That is, if A, B E MM(R), then A and B are similar if and only if the polynomial matrices XI,  A and X In  B are equivalent in Mn (R[X ]) . The proof we have given for Theorem 1.7 goes through with no essential modifications. With this in mind, a consequence of Example 3.8 is that the polynomial matrix
X13A=
X p 0 X 1 0
E M3(R[X])
X is not equivalent to a diagonal matrix. This is clear since the proof of Proposition 3.7 would show that A and At were similar if XI3  A was equivalent to a diagonal matrix. 0
0
What this suggests is that the theory of equivalence for matrices with
entries in a ring that is not a PID (e.g., R[X] when R is a PID that is not a field) is not so simple as the theory of invariant factors. Thus, while Theorem 1.7 (extended to A E M,,(R)) translates the problem of similarity of matrices in M,(R) into the problem of equivalence of matrices in M,(R[X]), this merely replaces one difficult problem with another that is equally difficult, except in the fortuitous case of R = F a field, in which case the problem of equivalence in M,,(F[X]) is relatively easy to handle.
The invariant factors of a matrix A E M,n,n (R) (R a PID) can be computed from the determinantal divisors of A. Recall (see the discussion prior to Definition 4.2.20) that Q,,,,,, denotes the set of all sequences a = (i1, ... , ip) of p integers with 1 < it < i2 < < ip m. If a E Qp,,,,,, ,3 E Qj,,,, and A E M,n...n(R), then A[a 10] denotes the submatrix of A whose row indices are in a and whose column indices are in 0. Also recall (Definition 4.2.20) that the determinantal rank of A, denoted Drank(A), is the largest t such that there is a submatrix A[a 1,31 (where a E Qt,,,, and /3 E Qt,,,) with det A[a 10] 0. Since R is an integral domain, all the ranks of a matrix are the same, so we will write rank(A) for this common number. For convenience, we will repeat the following definition (see Definition 4.2.21):
:3.10) Definition. Let R be a PID, let A E M,n,,, (R), and let k be an integer
such that 1 < k _< min{m, n}. If det A[a
1
a] = 0 for all a E Qk,m,
l3 E Qk,n, then we set dk (A) = 0. Otherwise, we set dk(A) = gcd{det A[a 10] : a E Qk,m., 0 E Qk,n}. dk (A) is called the kth determinantal divisor of A. For convenience in some formulas, we set do(A) = 1.
(3.11) Lemma. Let R be a PID and let A, B E Mm,n(R). Suppose that A is equivalent ofdk(B) to B and that 1 < k < min{m, n}. Then dk(A) is an associate
5.3 Smith Normal Form
313
Proof. Suppose UAV = B where U E GL(m, R) and V E GL(n, R). If a E Qk,,n and 0 E Qk,n, then the CauchyBinet theorem (Thei.rem 4.2.34) shows that (3.7)
det B[a I p] _ E det U[a I w] det A[w I r] det V [r I p]. WEQk,,, TEQk.n
Thus, if dk(A) = 0 then det B[a I /3] = 0 for all a E Qk,n, Q E Qk,n and hence dk(B) = 0. If dk(A) 34 0 then dk(A) I detA[w I r] for all w E Qk,,n, r E Qk.n, so Equation (3.7) shows that dk(A) I det B[a I ,o] for all a E Qk,m, (3 E Qk,n Therefore, dk(A) I dk(B).
Since it is also true that A = U'BV1, we conclude that dk(A) = 0 if and only if dk(B) = 0 and if dk(A) 54 0 then dk(A) I dk(B)
and
dk(B) I dk(A)
0
so dk(A) and dk(B) are associates.
Now suppose that R is a PID and that A E Mm,n(R) is in Smith normal form. That is, we suppose that rank(A) = r and
A[0
Dr
0
0
where Dr = diag(al, ... ,Sr) with s; 36 0 (1 < i < r) and s,
I
s;+l for
10. By Theorem 5.7 (VT : Vs)2 < (VT : VT)(VS
VS)
Thus,
(VT:Vs)2=(VT:VT)(VS:V5) Since £(VT) = £(V5) = n, Corollary 5.8 then shows that VT modules. Hence T and S are similar.
Vs as F[X]0
(5.14) Remark. The restriction that F be algebraically closed in Theorem 5.13 is not necessary. Indeed, let K be an algebraically closed field containing F (see Remark 2.4.18 (3)) and consider A = [SIB and B = [T]B E Then A is similar to B in M,,(F) if and only if A is similar to B in M,,(K) (Theorem 3.15) and the rank condition in Theorem 5.13 does not depend upon which field we are using.
The computation of HomR(M, N) where M and N are finitely generated torsion Rmodules over a PID R (Lemma 5.3) is also useful for some applications other than Theorem 5.13. We will give one such example. Suppose that V is a finitedimensional vector space over a field F and T E EndF (V ). The centralizer of T in the ring EndF (V) is
C(T) = {U E EndF(V) : TU = UT}. Note that, according to Proposition 4.4.1, C(T) = EndF(x)(VT).
The Falgebra generated by T, namely, F[T], is certainly contained in the centralizer C(T). There is a theorem of Frobenius, which computes the dimension of C(T) over F. This result is an easy corollary of Lemma 5.3.
336
Chapter 5. Matrices over PIDs
(5.15) Theorem. (Frobenius) Let F be a field, V a finitedimensional vector
space over F, andT E EndF(V). Iff1(X), , fk(X) (where fi(X) divides f,+1(X) for 1 < i < k  1) are the invariant factors of T, then k
dimFC(T) = E(2k  2i + 1) deg(fi(X )).
(5.13)
,=1
Proof. By Lemma 5.3, 25
k
k
®®F[X]/(f,(X), fj(X)).
C(T) = EndF(XI(VT)
i=1 j=1
But (fi(X ), fj (X )) = (fmin{,, j} (X )), so k
dimF C(T) = E deg fmin{,,j}(X ) +.j=1
But
{(i, j) : 1 < i, j < k and
min{i, j} = t}] = 2k  2t + 1,
so k
dimF C(T) = E(2k  2t + 1) deg ft(X) t=1
as required.
We have observed above that F[T]C C(T). As a corollary of Frobenius's theorem, there is a simple criterion for when they are equal, i.e., a criterion for when every linear transformation that commutes with T is a polynomial in T.
(5.16) Corollary. Let T E EndF(V). Then F[T] = C(T) if and only if mT(X) = cT(X), i.e., if and only if VT is a cyclic F[X]module.
Proof. First note that dimFF[T] = degmT(X) and if {fi(X)} 1 are the invariant factors of T, then mT(X) = fk(X). By Equation (5.13) k
dimF C(T) =
(2k  2i + 1) deg fi(X)
,cl k1
= dimF F[T] + E(2k  2i + 1) deg fi(X ). ,=1
From this we see that C(T) = F[T] if and only if k = 1, i.e., if and only if VT is a cyclic F[X]module.
5.6 Exercises
337
(5.17) Corollary. If T E EndF(V) then dimF C(T) > n = dimF(V).
Proof. Since f 1 f;(X) = cT(X), it follows that k
k
dimF C(T) = 1:(2k  2i + 1) deg f; (X) >
deg f; (X) = n. 11
(5.18) Example. The Jordan matrix Ja,
n is cyclic, so
C(JA, n) = F[JA, n].
It is easily checked that a basis of F(JA, n] consists of the n matrices A; _ Jo, n for 0 < i < n  1. That is, a matrix A = (a.;j] commutes with J,\, n if and only if it is upper triangular and constant on the lines parallel to the main diagonal.
5.6 Exercises 1.
Compute a finite free presentation of the Zmodule
M=Z2®Z3o®Z. 2.
3.
Compute two distinct finite free presentations of the R(X module (P3)D, where P3 denotes the real vector space of polynomials of degree at most 3 and D E EndR(P3) is the differentiation map. Let M be an abelian group with three generators v1, v2i and v3, subject to the relations
2v1  4v2  2v3 = 0 10v1  6v2 + 4v3 = 0 6v1  12v2  6v3 = 0. Assuming the matrix identity 10 11 l
1 21
4. 5.
°1
[2
6, [0 0 1 ] _ [0 0 0]
show that M ?5 Z2 ® Z14 ® Z, and find new generators Wi, w2, and W3 such that 2w1 = 0, 14w2 = 0, and W3 has infinite order. Use Theorem 3.6.16 to give an alternative proof of Theorem 2.2.
Construct a matrix A E GL(4, Z) with
rowl(A)=(12 10 6.
Construct a matrix A E GL(3, Q(X)) with
9
8).
Chapter 5. Matrices over PIDs
338
coll(A) = [X(X  1) X2 7.
X+1]'.
Construct a matrix A E M3(Z[i]) with
row2(A) _ [12i 1+3i 3i] 8.
and with det A = 2 + i. Reduce each of the following matrices to Hermite normal form: 2 6 91 (a) A 2 0 4 1 E M3(Z). 2
1
1
(b) B = [a 0X 1 + X 22X
E M3(Q[X]).
L
(c) C= 9.
3+i] EM2(Z[i]).
17i
Write the unimodular matrix
[3
A= 3 6
1
31 3 EJY13(Z)
3
5
3
as a product of elementary matrices. 10.
Let R = Z[/]. Show that no matrix [ a d ] with a = 2 and c = 1
is left equivalent to a matrix in Hermite normal form. Same as Exercise 10 with R = Q[X2, X3] and a = X2, C = X3. Let R be a PID. Show that there is a onetoone correspondence between the left ideals of M (R) and the Rsubmodules of Ml,,,(R). 13. Find the Smith normal form for each of the following matrices: 11. 12.
2
(a)
0 2
6
142
14
8
(d) 14.
0 0
E M3(Z).
6 8
X(X  1)3 (c)
E M3(Z).
2
1
(b)
10
0
1 3 4
0
0l
0
XJ
(X  1)
1+8i 23 + 2i
5+i
13i
E
0
E M 2(Z[i]).
Find the invariant factors and elementary divisors of each of the following matrices: (a)
0 0
2+4X
0
4+X 5
76X 2X
E 1113(ZS[X]).
0
(b) diag(20, 18, 75, 42) E 1i14(Z). 15.
(c) diag (X(X  1)2, X(X  1)3, (X  1), X). Let R be a PID and let A E Mm,,,(R) with m < n. Extend Theorem 2.2 by proving that there is a matrix B = [A, ] E (so that Al E
5.6 Exercises
339
(R)) such that det B is an associate of d..(A), the m`h determinantal M,, divisor of A. (Hint: First put A in Smith normal form.) 16.
Let S = {vi, ... , vk } C M",1(R where R is a PID. Show that S can be extended to a basis of M",1(R) if and only if dk(A) = 1, where A= [vi ... Vk ].
17.
Suppose that A E M3(Z) and det A = 210. Compute the Smith normal form of A. More generally, suppose that R is a PID and A E M"(R) is a matrix such that det A is squarefree. Then compute the Smith normal form of A. Let A E M" (Z) and assume that det A y6 0. Then the inverse of A exists in M"(Q), and by multiplying by a common denominator t of all the nonzero
18.
entries of A, we find that to1 E M"(Z). Show that the least positive integer t such that tA'1 E M"(Z) is t = Is"(A)l where s"(A) is the n1h invariant factor of A. 19.
Let A, B E M"(Z) such that AB = kI" for some k 54 0. Show that the
invariant factors of A are divisors of k. Let R be a PID and let A E M"(R), B E M,"(R). Show that the elementary divisors of A®B are the product of elementary divisors of A and of B. More precisely, if p' is an elementary divisor of A® B where p E R is a prime, then pr = pkpl where pk is an elementary divisor of A and p' is an elementary divisor of B, and conversely, if pk is an elementary divisor of A and p' is an elementary divisor of B, then pk+i is an elementary divisor of A 0 B. 21. Let A E M4(F) where F is a field. If A has an invariant factor s(X) of degree 20.
2 show that the Smith normal form of XI4  A is diag(1, 1, s( X), s(X)). Conclude that CA(X) is a perfect square in F[X). 22. Find all integral solutions to the following systems AX = B of equations:
23.
0
2
1
1
1
2
0
0 1
(c) A= [68
19
(b) A=
14
[51
B
,
1
7
B= [7]'
22]'
Show that the matrices A = [ 8
i ] and B = [ 232 15 ] in M2 (Q) are
similar.
24. Show that the matrices
25.
(0)
10
3
4
021
and
0
2
4
5
0
1
0
01
1
[0
0
are similar in M3(Z7). Show that the matrices 1]
and
1
[1
26.
0
0,
1
are similar in M3(Z3). Find the characteristic polynomial, invariant factors, elementary divisors, rational canonical form, and Jordan canonical form (when possible) of each of the matrices from Exercise 73 of Chapter 4. Additionally, find bases of Q" with respect to which the matrix (or linear transformation) is in rational
or Jordan canonical form. Do this exercise by reducing XI"  A to Smith
Chapter 5. Matrices over PIN
340
canonical form and compare your results with the same calculations done in Chapter 4. 27.
Find an example of a unimodular matrix A E M3(Z) such that A is not similar to A'. (Compare with Example 3.8.)
28.
Show that the matrix A =
2X X I is not equivalent in M2(Z(XJ) to a
2
diagonal matrix. (Hint: Use `Fitting ideals.) 29.
Let Z" have the standard basis lei, ... ,e"} and let K C Z" be the submodule generated by f, _ j a,jej where a,j E Z and 1 < i < n. Let A = [a,j] E MM(Z) and let d = det A. Show that Z/K is torsion if and only
ifdetA=d00andifd#0showthatIZ/Kl=l 1.
Suppose that an abelian group G has generators xl, x2, and x3 subject to the relations xl  3x3 = 0 and xl + 2x2 + 5x3 = 0. Determine the invariant factors of G and Cl iif G is finite. 31. Suppose that an abelian group G has generators xi, x2, and x3 subject to the relations 2x1  x2 = 0, xl  3x2 = 0, and x1 + x2 + x3 = 0. Determine the invariant factors of G and Cl lif G is finite. 30.
32. 33.
Verify the claim of Example 5.1.
Let F be a field and let A E M"(F), B E Mm(F). Show that the matrix equation
AX  XB = 0 for X E M",,"(F) has only the trivial solution X = 0 if and only if the
characteristic polynomials cA(X) and CB(X) are relatively prime in FIX]. In particular, if F is algebraically closed, this equation has only the trivial solution if and only if A and B have no eigenvalues in common. 34. Let F be a field. Suppose that A = Al ® A2 E M"(F) where Al E Mk(F) and A2 E Mm(F) and assume that cA,(X and CA3(.X) are relatively prime.
Prove that if B E M"(F) commutes with A, then B is also a direct sum
35.
B = B1 ® B2 where B1 E Mk(F) and B2 E M,,,(F). Let F be a field. Recall that C(f(X)) denotes the companion matrix of the monic polynomial f (X) E FIX]. If deg(f (X)) = n and deg(g(X)) = n, show
that
rank(C(f (X)) ® In  I" ® C(9(X ))) = deg(lcm{ f (X), 9(X)}). 36.
Let V be a finitedimensional vector space over a field F and let T E EndF(V). Prove that the center of C(T) is FIT].
Chapter 6
Bilinear and Quadratic Forms
6.1 Duality Recall that if R is a commutative ring, then HomR(M, N) denotes the set of all Rmodule homomorphisms from M to N. It has the structure of an Rmodule by means of the operations (f + g) (x) = f (x) + g(x) and (a!) (x) = a(f (x)) for all x E M, a E R. Moreover, if M = N then HomR(M, M) =
EndR(M) is a ring under the multiplication (fg)(x) = f (g(x)). An Rmodule A, which is also a ring, is called an Ralgebra if it satisfies the extra axiom a(xy) = (ax)y = x(ay) for all x, y E A and a E R. Thus EndR(M) is an Ralgebra. Recall (Theorem 3.4.11) that if M and N are finitely generated free Rmodules (R a commutative ring) of rank m and n respectively, then HomR(M, N) is a free Rmodule of rank mn.
In this section R will always denote a commutative ring so that HomR(M, N) will always have the structure of an Rmodule. (1.1) Definition. If M is an Rmodule, then HomR(M, R) is called the dual module of M and is denoted M*.
(1.2) Remark. If M if free of rank n then M' = HomR(M, R) R^ °5 M by Corollary 3.4.10. Note, however, that this isomorphism is obtained by choosing a basis of M' and M. One particular choice of basis for M* is the following, which is that described in the proof of Theorem 3.4.11 if the basis 111 is chosen for R. (1.3) Definition. If M is a free Rmodule and B = {v1, ... , vn} is a basis
of M, then the dual basis of M' is defined by B' = {vi, ... , v;} where vi E M' is defined by
v,(vi)=bij=
1
0
ifi=j
ifi#j.
(1.4) Example. Let R = Z and M = V. Consider the basis B = {vl = (1, 0), v2 = (0,1)}. Then vi (a, b) = a and vz (a, b) = b. Now consider the
Chapter 6. Bilinear and Quadratic Forms
342
basis C = {wl = (1, 1), w2 = (1, 2)}. Then (a, b) = (2a  b)wl + (b  a)w2 so that wi (a, b) = 2a  b and w2 (a, b) = b  a. Therefore, vl 34 w' and v2 # w2. Moreover, if D = Jul = (1,0), u2 = (1,1)} then ui(a,b) = a  b and u2 (a, b) = b so that ui 54 vl even though ul = vl. The point is that an element v, in a dual basis depends on the entire basis and not just the single element vi.
(1.5) Proposition. Let M be a free Rmodule of finite rank n and let B =
{v1,...,v } be a basis of M. Then B'={vl,...,vn} is a basis of M'. Proof. B' is the basis produced in the proof of Theorem 3.4.11.
0
(1.6) Corollary. Let M be a free Rmodule of finite rank n and let B = {vl, ... vn} be a basis of M. Then the map w : M i M' defined by n
n
waivi) _ i.1 aivi i=1 is an Rmodule isomorphism. Proof.
O
The isomorphism given in Corollary 1.6 depends upon the choice of a basis of M. However, if we consider the double dual of M, the situation is much more intrinsic, i.e., it does not depend upon a choice of basis. Let M be any Rmodule. Then define the double dual of M, denoted M",
by M" = (M')' = HomR(M', R). There is a natural homomorphism n: M M" = HomR(M', R) defined by 17(v)(w) = w(v)
for allvEMandWEM'=HomR(M,R). (1.7) Theorem. If M is a free Rmodule, then the map r, : M y M" is injective. If rank(M) < oo then r) is an isomorphism.
Proof. Suppose that v # 0 E M. Let B be a basis of M. If v = alvl + + anvn where al # 0 and {v1i ... , vn} C B, then we can define an element w E M' by w(vi) = 1 and w(w) = 0 for all w 34 vl E B. Then + anvn) = al # 0. Hence, n is injective. Now suppose that rank(M) < oo and let B = {vl, ... ,vn} be a basis
7](v)(w) = w(alvl +
of M. Let B' = {vi, ... ,v;a} be the dual basis of M' and let B" = {vi', ... , vn } be the basis of M" dual to the basis B' of M'. We claim that i(vi) = v1' for 1 < i < n. To see this, note that rl(vi)(vv) = viNvi) = bi j
= v,' (vj*)
6.1 Duality
343
Since 77(v1) and v;' agree on a basis of M', they are equal. Hence, q(M) (vl', ... vn) = M" so that q is surjective, and hence, is an isomorphism. 0
(1.8) Remark. For general Rmodules M, the map q : M  M" need not be either injective or surjective. (See Example 1.9 below.) When q happens to be an isomorphism, the Rmodule M is said to be reflexive. According to Theorem 1.7, free Rmodules of finite rank are reflexive. We shall prove below that finitely generated projective modules are also reflexive, but first some examples of nonrefiexive modules are presented. (1.9) Examples.
(1) Let R be a PID that is not a field and let M be any finitely generated nonzero torsion module over R. Then according to Exercise 9 of Chap
ter 3, M' = HomR(M, R) = (0). Thus, M" = (0) and the natural map q : M ' M" is clearly not injective. (2) Let R = Q, and let M = ®nENQ be a vector space over Q of countably infinite dimension. Then M' °' fl EN Q. Since
M=®QC fQ, nEN
nEN
we see that M' 5 M ® M' where M' is a vector space complement of M = ®nENQ in M. Then
M" = M. ®(M')' so that M" contains a subspace isomorphic to M. But ®nENQ is countably infinite, while the infinite product fl EN Q " M' is un
countable (the decimal representation identifies every real number with an element of 11nEN Q). Therefore, q : M  M" cannot be surjective by cardinality consideration and we conclude that M is not a reflexive Qmodule. Let Ml and M2 be Rmodules and let M = Ml ®M2. Then according to Corollary 3.3.13, M" °' Ml** ED M2**. In order to study reflexivity for direct sums and summands, it is necessary to identify carefully this isomorphism. To this end, define W : (MI ®M2)"  Ml** ED M2
by 'Y(w) = (w;, w2), where w, E M;' is defined by w,(9;) = w(9, o ir;) for each 0, E M, = HomR(M;, R). 7r; : Ml ® M2 + M, is the canonical projection map. Similarly, define OD: M'
M2' i (Ml
®M2).'
Chapter 6. Bilinear and Quadratic Forms
344 by
4)(wi, w2)(8) =w1(Oot1)+w2(Oot2)
where 0 E (M1 ® M2)' and i, : Mi + M1 ® M2 is the canonical injection. (1.10) Lemma. 1Y and 4) are inverse Rmodule homomorphisms.
Proof. Let w E (M1 ®M2)" and let 0 E (M1 (D M2)'. Then 0 o T(w)(0) = 4)(wi, w2)(0)
=w1(00ll)+w2(00t2) =w(9ot1 0711)+w(B01.207r2) =w(Ootl o7r1 +001.20712) =w(Oo(t1o7r1+1.20712)) =w(Oo1M1®M2)
= w(e), and
('I+ 0 4)(wl, w2))(01, 02) ='I'(Nwl, w2))(01, 02) = (4)(wl,w2)(01 0 711), 4)(wl,w2)(02 0 7r2)) = (w1(01 0711 0ti),w2(02071201.2)) _ (w1(0i),w2(02))
_ (wl,w2)(01,02).
Therefore, 4)o 41 = 1(MI(DM2).. and
W o 4) = 1Mi ®Mr,
0
and the lemma is proved.
Now let 71i:Mi'M;' (i = 1, 2) andq :M1ED M2 (M1(D M2)" be the natural maps into the double duals. (1.11) Lemma. Using the notation introduced above, there is a commutative diagram M1 ®M2
InM2)..
(M1 (D
w
That is, W 077 = (711, 712)
M1"eMM'
6.1 Duality
345
Proof. ((Y
rl) (v1, v2)) (wl, W2) =' (77(v1, v2)) (w1, w2)
= (rl(vl, v2)(w1 o rrl), rl(vl, v2)(w2 o l2)) = ((w1 o 7r1)(vl, v2) , (w2 o rr2)(vl, v2)) = (w1(vI), w2(v2))
= ((rll, r72)(v1, V2)) (w1, L02)
That is,Worl=(771,772). (1.12) Lemma. (1) 11 : Ml ®M2 p (MI ® M2)** is injective if and only if rli
: M; + Mi** is injective for each i = 1, 2. (2) MI ® M2 is reflexive if and only if MI and M2 are reflexive.
Proof. Both results are immediate from Lemma 1.11 and the fact that IF is an isomorphism.
(1.13) Proposition. If P is a projective Rmodule, then 77 : P > P** is injective. If P is also finitely generated, then P is reflexive. Proof. Since P is projective, there is an Rmodule P' such that P ® P' = F where F is a free Rmodule (Theorem 3.5.1), and furthermore, if P is finitely generated, then F may be taken to have finite rank (Corollary 3.5.5). The result now follows from Lemma 1.12 and Theorem 1.7.
The remainder of this section will be concerned with the relationship between submodules of an Rmodule M and submodules of the dual module
M*. The best results are obtained when the ring R is a PID, and the module M is a finite rank free Rmodule. Thus, we will make the following convention for the rest of the current section.
Convention. For the remainder of this section R will denote a PID and M will denote a free Rmodule of finite rank unless explicitly stated otherwise. (1.14) Definition. (1) If N is a submodule of M, then we define the hull of N, denoted
Hull(N)={x'EM:rx'EN for somer540ER}. If A is a subset of M, then we define Hull(A) = Hull((A)). (2) If A is a subset of M then define the annihilator of A to be the following
subset of the dual module M*:
K(A) = Ann(A)
={wEM*:Ker(w)DA}
={wEM*w(x)=0 for all xEAl C M*.
346
(3)
Chapter 6. Bilinear and Quadratic Forms
If B is a subset of M' then define the annihilator of B to be the following subset of M:
K'(B) = Ann(B)
={xEM:w(x)=O forallwEB} C M.
(1.15) Remarks. (1) If N is a submodule of M, then M/ Hull(N) is torsionfree, so Hull(N)
is always a complemented submodule (see Proposition 3.8.2); further
more, Hull(N) = N if and only M/N is torsionfree, i.e., N itself is complemented. In particular, if R is a field then Hull(N) = N for all subspaces of the vector space M. (2) If A is a subset of M, then the annihilator of A in the current context
of duality, should not be confused with the annihilator of A as an ideal of R (see Definition 3.2.13). In fact, since M is a free Rmodule and R is a PID, the ideal theoretic annihilator of any subset of M is automatically (0).
(3) Note that Ann(A) = Ann(Hull(A)). To see this note that w(ax') _ 0 r* aw(x') = 0. But R has no zero divisors, so aw(x') = 0 if and only if w(x') = 0. Also note that Ann(A) is a complemented submodule
of M' for the same reason. Namely, aw(x) = 0 for all x E A and
a#OER *w(x)=0 forallxEA.
(4)
Similarly, Ann(B) = Ann(Hull(B)) and Ann(B) is a complemented submodule of M for any subset B C M*.
The concepts of annihilators of subsets of M and M' will be used to get a duality between submodules of M and M'. But since annihilators of subsets are complemented submodules, we see immediately that it is necessary to restrict any correspondence between submodules of M and M* to the set of complemented submodules. Thus, if M if a free Rmodule, then we will denote the set of all complemented submodules by C(M). The following result is a collection of straightforward properties of annihilators. The verifications are left as an exercise.
(1.16) Proposition. Let M be a free Rmodule of finite rank, let A, A1, and
A2 be subsets of M, and let B, B1, and B2 be subsets of M. Then the following properties of annihilators are valid: (1) If Al C A2, then K(A1) J K(A2).
(2) K(A) = K(Hull(A)). (3) K(A) E C(M'). (4) K({0}) = M' and K(M) = {0}.
6.1 Duality
(5) K'(K(A))
347
A.
(1') If B1 C B2, then K'(B1) D K'(B2).
(2') K'(B) = K'(Hull(B)). (3') KO(B) E C(M).
(4') K'({O}) = M and K'(M') = {0}. (5') K(K'(B)) B. 0
Proof. Exercise.
The following result is true for any reflexive Rmodule (and not just finite rank free modules). Since the work is the same, we will state it in that context:
(1.17) Lemma. Let M be a reflexive Rmodule and let n : M  M" be the natural isomorphism. Then for every submodule T of M', we have
r)(K'(T)) = K(T) C M. Proof. Let w E K(T). Then w = rl(x) for some x E M. For any t E T,
t(x) = zl(x)(t) = w(t) = 0 because w E K(T). Therefore, x E K'(T) by definition and hence K(T) C i7(K*(T)) Conversely, if x E K' (T) then 0 = t(x) = 77(x)(t)
for any t E T so r)(x) E K(T) by definition. Thus, rt(K'(T)) C K(T), and the lemma is proved.
(1.18) Theorem. Let M be a free Rmodule of finite rank, let S be a complemented submodule of M, and let T be a complemented submodule of M*. Then
rank(M) = rank(S) + rank(K(S)), and
rank(M') = rank(T) +rank(K'(T)). Proof. Let B1 = {v1i ... ,vk} be a basis of S. Since S is complemented, it follows (Corollary 3.8.4) that 81 extends to a basis
B = {vl, ... , Vk) Vk+l, ... , vm} of M. Let B' = {v,*, ... , vm } be the basis of M dual to B. If i < k and j > k then vj*(vi) = 0. Therefore,
348
Chapter 6. Bilinear and Quadratic Forms
(vk+1, ... , v.n) C K(S). If w E K(S), then we may write w =
1 ajv,*, and if 1 < i < k, then
m
0 = W(v8) = E ajvj* (vi) = a,. j=1
Therefore, w = F jk+1 aj vj*, and hence, K(S) = (vb+1, ... , v;R)
so that rank(K(S)) = m  k = rank(M)  rank(S). Similarly,
rank(M') = rank(T) + rank(K(T)) = rank(T) + rank(ri(K'(T))) = rank(T) + rank(K'(T))
where the last equality is valid because q : M M" is an isomorphism, and hence, it preserves ranks of submodules. (1.19) Theorem. Let M be a free Rmodule of finite rank. Then the function
K : C(M)  C(M') is a onetoone correspondence with inverse K*.
Proof. We claim that for every complemented submodule S C M and T C M', we have
K'(K(S)) = S and
K(K'(T)) = T. We will prove the first of these equalities; the second is similar. First note the K'(K(S)) 2 S for every complemented submodule S C M by Proposition 1.16 (5), so Corollary 3.8.5 implies that it suffices to show
that rank(K'(K(S))) = rank(S). But rank(S) = rank(M)  rank(K(S)) and
rank(K(S)) = rank(M')  rank(K'(K(S))) by Theorem 1.18. Since rank(M) = rank(M'), the result follows.
6.1 Duality
349
(1.20) Definition. If M and N are Rmodules and f E HomR(M, N) then
the adjoint off is the function f' : N'  M' defined by f' (w) = w o f , that is, (f'(w))(x) = w(f(x))
forallxEM. (1.21) Remarks. (1)
f' : N'
M' is an Rmodule homomorphism.
(2) Ad : HomR(M, N)  HomR(N', M'), defined by Ad(f) = f', is an Rmodule homomorphism. (3) If M and N are free, Ker(f) is always a complemented submodule of
M, but Im(f) need not be complemented. (See Proposition 3.8.7.)
(1.22) Theorem. Let M and N be free Rmodules of finite rank and let f E HomR(M, N). Then
(1) Ann(Im(f )) = Ker(f') C N', (2) rank(Im(f')) = rank(Im(f))), and (3) Im(f') = Ann(Ker(f)) C M' if Im(f') is a complemented submodule of M*.
Proof. (1) Let w E N'. Then
w E Ker(f') p f'(w) = 0
qwo f =0 aw(f(x))=0 tlxEM aw(y)=0 VyEIm(f) a w E Ann(Im(f)). (2) Since f' : N' + M', Proposition 3.8.8 gives rank(N') = rank(Im(f')) + rank(Ker(f')) while Theorem 1.18 shows
rank(N) = rank(Im(f)) + rank(Ann(Im(f))). Since rank(N) = rank(N'), (2) follows from (1). (3) Now let r E M'. Then r E Im(f') a r= f' (w) for some w E N'
a r(x) = w(f (x)) Vx E M. If x E Ker(f) then f (x) = 0, so w(f (x)) = 0. Therefore, r(x) = 0, and we conclude that r E Ann(Ker(f )). Hence, Im(f') C Ann(Ker(f )). By Theorem 1.18 and part (2), rank(Ann(Ker(f))) = rank(M)  rank(Ker(f)) = rank(Im(f)) = rank(Im(f')). Since Im(f') is assumed to be complemented, we conclude that Im(f') _ Ann(Ker(f)). 0
Chapter 6. Bilinear and Quadratic Forms
350
(1.23) Corollary. Let F be a field, let V and W be finitedimensional vector spaces over F, and let f E HomF(V, W). Then
(1) f is injective if and only if f * is surjective; (2) f is surjective if and only if f * is injective; and (3) f is an isomorphism if and only if f * is an isomorphism. Proof.
(1.24) Proposition. Let M and N be free Rmodules of finite rank with bases B and C, respectively, and let f E HomR(M, N). Then [f"]C _ ([.f]CC)t
Proof. Let B = {v}1 and C = {wj}T 1. If A = [aid]
[f]' and B =
[bid] _ [f*]C., then by definition m
f (vj) = > akj wk k=1
and
f*
R
(wi) _ E bkivk k=1
But then aii = wi (.f(vj)) = (w: o f)(vj) = (.f*(w2M(vj) = bpi.
0
6.2 Bilinear and Sesquilinear Forms In this section we present an introduction to an important branch of mathematics that is the subject of much study. Throughout this section R will be a commutative ring with 1 and all Rmodules will be free. (2.1) Definition. A conjugation on R is a function c : R > R satisfying (1) (2) (3)
c(c(r)) = r for all r E R; c(rl + r2) = c(ri) + c(r2) for all ri, r2 E R; and c(rlr2) = c(rl)c(r2) for all r1, r2 E R.
6.2 Bilinear and Sesquilinear Forms
351
That is, a nontrivial conjugation of R is a ring automorphism, which has order 2 as an element of the group Aut(R). (2.2) Examples.
(1) Every ring has the trivial conjugation c(r) = r. Since Aut(Q) = {1Q), it follows that the trivial conjugation is the only one on Q. The same is true for the ring Z. (2) The field C has the conjugation c(z) = z, where the righthand side is complex conjugation. (This is where the name "conjugation" for a function c as above comes from.)
(where d is not a square) both have the conjugation c(a + bVd) = a  bf .
(3) The field Q[/] and the ring
Because of Example 2.2 (2), we will write 'r, instead of c(r), to denote conjugation.
(2.3) Definition. Let M be a free Rmodule. A bilinear form on M is a
function 0: M x M R satisfying (1) 4 (rlxl + r2x2, y) = r14 (xl, y) + r20(x2, y), and (2) O(x, rlyl +r2y2) = ri (x, yl) +r2j(x, y2) for all x1, x2, y1, y2 E M, and r1i r2 E R. A sesquilinear form on M is a function 0: M x M  R satisfying (1) and (2) O(x, r1y1 + r2y2) = f1 (x, yl) + r2O(x, y2)
for a nontrivial conjugation r
T on R.
Observe that this notion is a generalization of the notion of inner product space that we considered in Section 4.6. Some (but not all) authors use
the term "inner product space" to refer to this more general situation. (Strictly speaking, in the second part of the definition we should say that 0 is sesquilinear with respect to the given conjugation, but we shall assume that we have chosen a particular conjugation and use it throughout.)
(2.4) Definition. Let R be a ring with conjugation and let M and N be Rmodules. A map f : M + N is called an antihomomorphism if f(r1m1 +r2m2) = T1f(m1) + T2f(m2) for all r1, r2
R, m1, m2
M.
We observed in Section 6.1 that there is no canonical isomorphism from M to its dual module M'; however, a bilinear form produces for us a canonical map, and, conversely, a map produces a canonical form. (Here we
352
Chapter 6. Bilinear and Quadratic Forms
do not necessarily have isomorphisms, but we shall investigate this point shortly.) (2.5) Proposition.
(1) Let 0 be a bilinear (reap., sesquilinear) form on M. Then am M  M', defined by am(y)(x) = O(x, y)
is an Rhomomorphism (reap., Rantihomomorphism).
(2) Let a : M + M' be an Rhomomorphism (neap., Rantihomomorphism). Then 0Q : M x M  R, defined by
ma(x, y) = a(y)(x) is a bilinear (reap., sesquilinear) form on M. Proof. Exercise.
(2.6) Examples.
(1) Fix s E R. Then ¢(rl, r2) = rlsr2 (reap., = rl02) is a bi (reap., sesqui) linear form on R.
(2) O(x, y) = xty is a bilinear form on Mn,l(R), and O(x, y) = x1V is a sesquilinear form on Mn,1(R). Note that ji is obtained from y by entry by entry conjugation. (3) More generally, for any A E Mn(R), O(x, y) = xtAy is a bilinear form, and O(x, y) = xtA'y is a sesquilinear form on Mn,l (R).
(4) Let M = Mn,,n(R). Then 4,(A, B) = Tr(AtB) (reap., ¢(A, B) _ TT(AtB)) is a bi (reap., sesqui) linear form on M. (5) Let M be the space of continuous real (reap., complex) valued functions on [0, 1]. Then
Of, 9) =
f
1
.f (x)9(x) dx
0
is a bilinear form on the R (reap., C) module M. If M is the space of continuous complexvalued functions on [0, 1], then
O(,f, 9) = fo f(x)9(x) dx 1
is a sesquilinear form on the Cmodule M.
We will often have occasion to state theorems that apply to both bilinear and sesquilinear forms. We thus, for convenience, adopt the language that 0 is a b/slinear form means 0 is a bilinear or sesquilinear form. Also,
6.2 Bilinear and Sesquilinear Forms
353
the theorems will often have a common proof for both cases. We will then write the proof for the sesquilinear case, from which the proof for the bilinear case follows by taking the conjugation to be trivial (i.e., r = r for all r E R). We will start our analysis by introducing the appropriate equivalence relation on b/slinear forms.
(2.7) Definition. Let 01 and 02 be b/slinear forms on free Rmodules M1 and M2 respectively. Then ¢1 and 02 are isometric if there is an Rmodule isomorphism f : M1 . M2 with 02 (f (X), f(y)) _ 01(x, y)
forall
x,yEM1.
(If there is no danger of confusion, we will call M1 and M2 isometric.) The map f is called an isometry.
Our object in this section will be to derive some general facts about b/slinear forms, to derive canonical forms for them, and to classify them up to isometry in favorable cases. Later on we will introduce the related notion of a quadratic form and investigate it. We begin by considering the matrix representation of a b/slinear form with respect to a given basis.
(2.8) Definition. Let M be a free Rmodule of rank n with basis B = {vi, ... , V.) and 0 a b/slinear form on M. Define the matrix of ¢ with respect to the basis B, denoted [01g, by entii (1018) = 0(vi, vi)
1[i) = det(O) # 0). Thus, unless char(R) = 2 and 0 is symmetric bilinear, we are done. If char(R) = 2 and 0 is symmetric bilinear, we cannot apply induction yet as we do not know that the form t/, is odd. Indeed, it is possible that t/i is even and so there is more work to be done. First, consider the case rank(M) = 2. Let a = O(x, x) and choose a basis {x, x2} of M. Then, in this basis has the matrix
[b c, with a 54 0. Let e = b/a. Then [ab
1
c] [0 1]
Ie
 [0
d,
1) (with d = ae2 + c) and 0 is diagonalized. Now suppose that rank(M) > 3. Find x as above with 0(x, x) = a 0 0
and write M = N I N1 as above. Pick y E N1. If 0(y, y) 0 0, then OINl is odd and we are done (because we can apply induction). Thus, suppose 0(y, y) = 0. Since VP = OIN1 is nonsingular, there is z E N1 with
0(y, z) = b # 0. If 0(z, z) 0 0, then ik is odd and we are again done by induction. Thus suppose 0(z, z) = 0. Let M1 be the subspace of M with basis {x, y, z} and let 01 = OIM, . Then, in this basis 01 has the matrix
A=
a
0
0
0
0
0
b
b 0
1
e
e
1
1
0
1
0
1
Let e = b/a and
P= and note that det(P) = 1. Then a
PAP =
0
0 be
(b + a)e
0
(b + a)e
be
0
in a basis, which we will simply denote by {x', y', z'}. Now let N be the subspace of M spanned by x', and so, as above, M = N 1 N1. But now 0 = OI N1 is odd. This is because y' E N1 and t,i(y', y') = be 0 0. Thus, we may apply induction and we are done. 0 (2.43) Example. We will diagonalize the symmetric bilinear form with matrix
6.2 Bilinear and Sesquilinear Forms
A=
0
1
1
0
2 3
2
3
0
367
over Q. The procedure is to apply a sequence of elementary row/column operations. If A is symmetric and E is any elementary matrix, then EtAE is also symmetric. We indicate the matrices and record the results. Al = T12(1)AT21(1) =
2
1
5
1
0
3
5
3
0
0
2
A2 = T21(1/2)A1T12(1/2) =
1/2
5
A3 = T31(5/2)A2T13(5/2) =
A4 = T32(1)A3T23(1) =
5
1/2 1/2
0
0
2
0
0
0 0
1/2
1/2
1/2
25/2
2
0
0
1/2
0 0
0
0
12
The reader should not be under the impression that, just because we have been able to diagonalize symmetric or Hermitian forms, we have been able to classify them. However, there are a number of important cases where we can achieve a classification. (2.44) Corollary. Let 0 be a nondegenerate symmetric bilinear form on a module M of finite rank over a field R of characteristic not equal to 2 in which every element is a square. (Note that R = C satisfies this hypothesis.) Then 0 is determined up to isometry by rank(M). If rank(M) = n then ¢ is isometric to nil]. Proof. This follows immediately from Theorem 2.42 and Remark 2.15.
0
(2.45) Corollary. Let 0 be a nondegenerate symmetric bilinear form on a module M of finite rank over a finite field R of odd characteristic. Then 0 is determined up to isometry by rank(M) and det(O), the latter being well defined up to multiplication by a square in R. Let x E R be any element
that is not a square, and let rank(M) = n. If det(4) is a square, then 0 is isometric to nil]. If det(O) is not a square, then 0 is isometric to (n  1)(1] 1 (x]. Proof. By Theorem 2.42 we know that 0 is isometric to the form (r1) 1 (r2] 1 ... 1 (rn]
368
Chapter 6. Bilinear and Quadratic Fbrms
for some ri E R. Note that the multiplicative group R' has even order, so the squares form a subgroup of index 2. Then det(4)) is a square or a nonsquare accordingly as there are an even or an odd number of nonsquares
among the {r,}. Thus, the theorem will be proved once we show that the form [ri] I [rj] with r, and rj both nonsquares is equivalent to the form [1] 1 (s] for some a (necessarily a square). ] in a basis B = {v1, V2} of M. R has an odd Thus, let [0]g = [
o°
number of elements, say 2k + 1, of which k + 1 are squares. Let A = {a2r1 : a E R} and B = {1b2r2 : b E R}. Then A and B both have k+1 elements,
soAf1B00.Thus, forsome ao,b0ER, ar 1
= 1  ba re,
4r, +b 0 r2 = 1.
Let N be the subspace of M spanned by a0v1 + bov2. Then 4)]N = [1] and
M = N 1 N L, so, in an appropriate basis, 0 has the matrix [ o , J, as 0
claimed.
(2.46) Corollary. Let 0 be a nondegenerate symmetric bilinear form on a module M of finite rank over a finite field R of characteristic 2. Then 0
is determined up to isometry by n = rank(M) and whether ¢ is even or odd. If n is odd, then 0 is odd and is isometric to n[l]. If n is even, then either 0 is odd and isometric to n[1], or 0 is even and 0 is isometric to
(n/2) l0
11.
Proof. Since R' has odd order, every element of this multiplicative group is a square. If 0 is odd then, by Theorem 2.42, 0 is diagonalizable and then 0 is isometric to n[1] as in Corollary 2.44. Suppose that 0 is even. Note that an even symmetric form over a field of characteristic 2 may be regarded as a skewsymmetric form. Then, by 0 Corollary 2.36, n is even and 0 is isometric to (n/2) [ ° o ] .
(2.47) Theorem. (Witt) Let 0 be a (+1)symmetric b/slinear form on a module M of finite rank over a field R. If char(R) = 2, assume also that 0 is Hermitian. Let N1 and N2 be submodules of M with 4)]N, and 4)IN2 non
singular and isometric to each other. Then OiN; and 4)IN; are isometric.
Proof. If N1 = N2, then Nl = N21, and there is nothing to prove; so we
assume N1 0 N2. Let 4i = ON,, and let f : N1  N2 be an isometry between 01 and 02. We prove the theorem by induction on n = rank(Ni) = rank(N2). Let n = 1. Let m = rank(M). If m = 1 the theorem is trivial. Let m = 2. Let v1 generate N1 and v2 = f (v1) generate N2, so
6.2 Bilinear and Sesquilinear Forms
369
((vl, v1) = 0(v2, v2) = a # 0.
Then M has bases Bi = {vi, v; } with vi E Nil, i = 1, 2, and hence JOJ,g, = diag(a, bi) for i = 1, 2, with b1 and b2 both nonzero. By Theorem 2.13, there is an invertible matrix P with Pt[0Js1P = 101s21
and taking determinants shows that ab2 = ccab1 (where c = det(P)); so g : NIL + N2' defined by 9(vi) = c1vs gives an isometry between cIN. and 0IN.L.
Next let m > 3 and consider the submodule N12 of M with basis {v1, v2}, where v1 generates N1, and v2 = f(v1) generates N2. Then 4,(v1, vi) = O(v2, v2) # 0.
(Since we are assuming Ni $ N2, v1 and v2 are linearly independent in M and so N12 has rank 2.) Consider 01 N, .. Either it is nonsingular or it is not. First suppose that 4,IN1, is nonsingular. Then, by the case m = 2 we have that 01N , IN12 and OIN2 nN1, are isometric, and hence, 0I(N;nN12)1Ns
and '0I(N1nN12)1N, L
are also isometric. But in this case M = N12 1 N12, from which it readily follows that
(N: nN12)1N 2=N; , yielding the theorem. Now suppose WI N1, is singular. Then there is a 0 0 w E N12 with ¢(v, w) = 0 for all v E N12. Suppose there is a v3 E M with q5(v3i w) 0 0. (Such an element v3 certainly exists if 0 is nonsingular on M.) Of course, v3 0 N12, so {v1, v2, v3} form a basis for a submodule N123 of M of rank
3 and 0IN1 is nonsingular. (To see this, consider the matrix of 0 in the basis {v1, w, v3} of N123.) Now for i = 1, 2,
N123=Ni1(Ni nN123) and w E Ni n N12 3 with 4,(w, w) = 0, so there is a basis {w, w1 } of Ni n N123 with 0 having matrix [° aj in that basis (with ai = di). We claim that any two such forms are isometric, and this will follow if, given
anyaERwitha=Z, there is abERwith Lb
1J 11
a110 1J  11 0J
If char(R) 96 2, take b = a/2 (and note that b = 6). If char(R) = 2, let c E R with c 34 c (which exists as we are assuming that 0 is Hermitian)
370
Chapter 6. Bilinear and Quadratic Forms
and let b = ac/(c+c). Hence, tINi nN,23 and 0IN2 nN,23 are isometric, and M = N123 1 N12 3 (as OI N, 2 3 is nonsingular); so, as in the case 01 N, 2 is nonsingular, it follows that 0IN. and OINz are isometric. It remains to deal with the case that O(v, w) = 0 for every v E M. We claim that in this case N1 = N2, so the theorem is true. To see this, let B1 2 = {V1, V21 and extend B1 2 to a basis B = {v1 i ... , vm } of M. Let A = [O]B. Then A[(1, 2)
1
(1, 2)] _
La
a,
with a 0 0 and a2  bb = 0, and we may assume w E N12 is given by
[w]B=[a b
0
...
0]t
(as w is well defined up to a scalar factor). Then 0(vi, w) = 0 for i = 3, m implies that there exist scalars ci so that [entli(A)1 = { (v1, vi)J = ci a ent2i(A) JJ
'(v2, vi)
,
i = 3, .
.
, m.
LbJ
If we let P be the matrix defined by
entii(P) = 1 ent2Z(P) = Ci
i = 3, ... , m
entij(P) = 0
otherwise,
then
PtAP = rA L0
0 *
(the righthand side being a block matrix). This is [¢]B' in the basis
13 ={v1i712,7/3,...,vm}
and then Nl = N2 is the subspace with basis {w, v3,
...
, vm}
This concludes the proof of the theorem in case n = 1. Now we apply induction on n. Assume the theorem is true for
rank(N1) = rank(N2) < n, and let rank(N1) = rank(N2) = n.
Let v1 E N1 with O(v1, v1) 0 0, and let v2 = &1) E N2; so cb(v2, v2) = O(711, 711) 54 0.
6.2 Bilinear and Sesquilinear Forms
371
Note that such an element exists by the proof of Theorem 2.42. Let N11 be the subspace of M generated by v1 and let N21 be the subspace of M generated by v2. Then
M=N111(NinN1)1NIL =N211(N,nN2)1N2. Then the case n = 1 of the theorem implies that
(NL1nN1)1Ni and (N21nN2)1N2 are isometric, and then the inductive hypothesis implies that NIL and N2 are isometric, proving the theorem.
This theorem is often known as Witt's cancellation theorem because of the following reformulation:
(2.48) Corollary. Let 01, 02, and 13 be forms on modules of finite rank over a field R, all three of which are either symmetric or Hermitian. If char(R) = 2, assume all three are Hermitian. If 01 is nonsingular and 01 1 02 and 01 1 03 are isometric, then 02 and 03 are isometric. Proof.
(2.49) Remark. Witt's theorem is false in the case we have excluded. Note
that 1
0
0
0
0
1
0
1
0
and
1
0
0
0
1
0
0
0
1
are isometric forms on (Fr2)3, as they are both odd and nonsingular, but I
0
1J
and
[01
1]
are not isometric on (F2)2, as the first is even and the second is odd. Now we come to the very important case of symmetric bilinear forms over R and Hermitian forms over C. (2.50) Definition. If 46 is a symmetric bilinear form over R, or a Hermitian
form over C, on a module M, then ¢ is said to be positive definite if ¢(v, v) > 0 for every v 34 0 E M. If q(v, v) < 0 for every v 36 0 E M, then 0 is said to be negative definite. (2.51) Theorem. (Sylvester's law of inertia) Let 0 be a nondegenerate sym
metric bilinear form over R = R, or Hermitian form over R = C, on a module M of finite rank over R. Then 0 is isometric to
372
Chapter 6. Bilinear and Quadratic Forms
r[1] 1 s[1] with r+s = n = rank(M). Furthermore, the integers r ands are well defined and 45 is determined up to isometry by rank(o) = n, and signature(O) = rs
Proof. Except for the fact that r and s are well defined, this is all a direct corollary of Theorem 2.42. (Any two of n, r, and s determine the third, and we could use any two of these to classify 0. However, these determine and are determined by the rank and signature, which are the usual invariants that are used.) Thus, we need to show that r and s are well defined by 0. To this end, let M+ be a subspace of M of largest dimension with OPM+ positive definite. We claim that r = rank(M+). Let B = {v1, ... be a basis of M with [0113 = Ir ®I,.
If Ml = (v1, ... , vr), then OIM, is positive definite. Thus, rank(M+) > r. This argument also shows that if M_ is a subspace of M of largest possible dimension with 01M_ negative definite, then rank(M_) > s.
We claim that r = rank(M+) and s = rank(M_). If not, then the above two inequalities imply
rank(M+) + rank(M_) > r + s = n, so M+f1M_ # {0}. Let x 96 0 E M+f1M_. Then 4(x, x) > 0 since x E M+, while 0(x, x) < 0 since x E M_, and this contradiction completes the proof. We present an alternative proof as an application of Witt's theorem. Suppose rl[1] I sl[1] and r2[1] 1 a2[ 1]
are isometric. We may assume rl c r2. Then by Witt's theorem, s1 [l) and (r2  rl)[1] 1 82[l] are isometric. As the first of these is negativedefinite, 0 so is the second, and so rl = r2 (and hence si = s2).
(2.52) Remark. If 0 is not assumed to be nondegenerate then applying Theorem 2.51 to the nondegenerate part of 0 (Definition 2.27) shows that a symmetric bilinear form over R = R or a Hermitian form over R = C is isometric to
r[1] 1 s[1] 1 k[0]
with r + s = rank(0) = rank(M)  k, and with r and a well defined by 0. Again, we let signature(46) =,r  s.
Of course, we have a procedure for determining the signature of 0, namely, diagonalize 0 and inspect the result. In view of the importance of this case, we give an easier method.
6.2 Bilinear and Sesquilinear Forms
373
(2.53) Proposition. Let 0 be a symmetric bilinear form over R = R, or a Hermitian form over R = C, on a module M of finite rank over R, and let A = [0]B for some basis B of M. Then
(1) rank(O) = rank(A); (2) all of the eigenvalues of A are real; and
(3) signature(O) = rs, where r (reap., s) is the number of positive (reap., negative) eigenvalues of A.
Proof. It is tempting, but wrong, to try to prove this as follows: The form 0 is diagonalizable, so just diagonalize it and inspect the diagonal entries. The mistake here is that to diagonalize the form 0 we take PAP, whereas to diagonalize A we take PAP', and these will usually be quite different. For an arbitrary matrix P there is no reason to suppose that the diagonal entries of P=AP are the eigenvalues of A, which are the diagonal entries of
PAP`. On the other hand, this false argument points the way to a correct argument: First note that we may write similarity as (P)'AP. Thus if P
is a matrix with P' = (P)'', then the matrix B = PtAP will have the same eigenvalues as A. Let us regard A as the matrix of a linear transformation a on R" where
R = R or R = C. Then A is either real symmetric or complex Hermitian. In other words, a is selfadjoint in the language of Definition 4.6.16. But then by Theorem 4.6.23, there is an orthonormal basis B = {v,, ... , v"} of R" with B = [a]B diagonal, i.e., P'AP = B where P is the matrix whose columns are v1, ..., v,,, and furthermore B E M"(R), where n = rank(A). But then the condition that B is orthonormal is exactly the condition that Pt = (15)1, and we are done. 0 There is an even handier method for computing the signature. Note that it does not apply to all cases, but when it does apply it is easy to use.
(2.54) Proposition. Let 0 be a symmetric bilinear form over R = R or a Hermitian form over R = C, on an Rmodule M of finite rank n. Let B be any basis for M and let A = [']B. Set bo(A) = 1, and for 1 < i < n, let 6i(A) = det(Ai), where Ai = A[(1, 2, ..., i) (1, 2, ..., i)] is the it' principal submatrix of A, i.e., the i x i submatrix in the upper lefthand corner.
(1) Ifbi(A)#0 foralli, then (2.7)
signature(O) = I{i : b;(A) and bi_,(A) have the same sign}l  I{i : bi(A) and bi_,(A) have opposite signs) .
(2) 0 is postive definite if and only if b; (A) > 0 for all i. (3) 0 is negative definite if and only if (1)ibi(A) > 0 for all i.
Chapter 6. Bilinear and Quadratic Forms
374
Proof We first prove part (1) by induction on n. The proposition is trivially
true if n = 1. Assume it is true if rank(M) < n, and consider M with rank(M) = n. Write Equation (2.7) as
signature(o) = r'(A)  s'(A).
(2.8)
Of course, r'(A) + s'(A) = n. Let B = {v1, ... , vn}, let N C M be the subspace with basis B' = (v1i ... ,vn_1), and let t = GIN. Set B = [,0jfj,. Then B = An1, so det(B) 54 0 by hypothesis, and thus ik is nonsingular. Let C' = {w1 i ... , wn_1 } be a basis of N with [01c' diagonal, say,
['G]c' = diag(c1, ... cn1) Then, by definition,
signature(o)_ I { i : c i > 0 } 1  {i:ci n  r. (2) If Ker(g1) = (0), then g cannot be written as a product of fewer than
n reflections.
Proof. (1) Let g = fy,
fy. and let Nj = Ker(fy,  1). Then
Since dim N) = n  1, it follows that (Proposition 3.8.10)
6.4 Exercises
391
n  r.
dim(Ni (2) This follows immediately from (1). (3.26) Corollary.
(1) If dim M = 2 then every isometry of determinant 1 is a reflection. (2) If dim M = 3 and g is an isometry of determinant 1, then g is the
product of 2 reflections. Proof. Exercise.
6.4 Exercises 1.
2.
Give an example of a field F with no/exactly one/more than one nontrivial conjugation. Let 0 be a b/slinear form (or 4i a quadratic form) on a free Rmodule N and
let f : M  N be an Rmodule homomorphism. Show that f'(0) defined by
(f * (4,))(x, y) = Of (x), f (y)),
or f' (4i), defined by
(f*(4))(x) = 4V W), 3. 4.
is a b/slinear (or quadratic) form on M. Let 0 be an even symmetric bilinear form over a PID R in which 2 is prime (e.g., R = Z) and suppose that det(4,) 2R. Show that rank(O) is even. Note that n 10 of is a form satisfying the conditions of Exercise 3, for any n. It is far from obvious that there are any other even symmetric nonsingular forms over Z, but there are. Here is a famous example. Consider the form 0 over Z with matrix 1
0
0
0
0
2
1
0
0
0 0
0
1
0
0
1
2
1
0
0
0
0 0
0
0
1
2
1
0
0
0
0
0
0
1
2
1
0
0
0
0
0
1
2
1
0 0
0
0
0
1
2
0
0 0
0 0
0
1
0
0
2
2
E8 =
5.
1
a Show that det(4,) = 1. b Show that signature(4,) = 8, where 0 is regarded as a form over R. a) Let 0 be a b /slinear form on a module M over a ring R. Suppose is an integral domain. If there is a subspace H C M with rank(H) > (1/2) rank(M), which is totally isotropic for 0, show that 0 is degenerate. b Suppose that 0 is either an even symmetric bilinear form on a module M over a ring R or a Hermitian form on a module M over a field R. If 0 is nonsin lar and there is a totally isotropic subspace H C M with rank(H) = (1/2) rank(M), show that 45 is isometric to
(rank(M)/2) [0
1 1
Chapter 6. Bilinear and Quadratic Forms
392 6.
Let 4; be a quadratic form associated to a bilinear form 0. Derive the following identities:
(a) 24D(x + y) = 24)(x, y) + 4)(x, x) + m(y, y)
(b) 4(x + y) + b(x  y) = 2($(x) + 4'(y)). The latter equation is known as the polarization identity. 7. Prove Proposition 2.5. 8. The following illustrates some of the behavior that occurs for forms that are not esymmetric. For simplicity, we take modules over a field F. Thus 4) will
denote a form on a module M over F. (a) Define the right/left/twosided kernel of 45 by
M°={yEM:0(x,y)=0 forallxEM} MM ={yEM:4)(y,x)=0 forallxEM} M°={yEM:4)(x,y)=m(y,x)=0 forallxEM}. Show that if rank(M) < oc, then M° = {0} if and only if M° = {0}.
Give a counterexample if rank(M) = oo. (b) Of course, M° = M° n Mi . Give an example of a form on M (with rank(M) < oo) where M° # {0}, Ml {0}, but M° = {0}. We say that 0 is right/left nonsingular if Mr = 0}/M1 = {0} and nonsingular if it is both right and left nonsingular. (c) If N C M, define
NN ={yEM:4)(x,y)=0 forallxEN} with an analogous definition for N1. Let N1 = Nr n NIL. Give an
example where these three subspaces of M are all distinct. (d) Suppose that N C M is a subspace and is nonsingular. Show that M = N ® N; and M = N ® N, L. Give an example where M # N ® N1. Indeed, give an example where N is a proper subspace of M but N1 = 101
9.
Let R be a PID, M a free Rmodule of rank 4, and let 0 be a nondegenerate
skewsymmetric form on M. Show that 0 is classified up to isometry by det(m) and by
gcd{4)(v, w) : v, w E M}.
Let R be a PID, and let 0 be a skewsymmetric form on a free Rmodule M. If det(4)) is squarefree, show that 0 is classified up to isometry by its rank. 11. Classify the following forms over Z (as in Theorem 2.35): 10.
(a)
(b)
12.
0
1
1
1
2
1
1 1
0
5
3
2
3
5
0
3
7
1
3
0
6
4 5
2
2
7
1
3
4
6 5
0
1
1
0
0 0 6 6 6 8
0
0 6 7 7
9
3
6 6 6 8 6 7 7 9 0 1 5 7 1
5 7
0 6 8
6 8 0 0
0 0
Find the signature of each of the following forms over R. Note that this also gives their diagonalization over R.
6.4 Exercises
40
40 50
9
12
5
(a)
2
(c)
4 6
9 12 3
4 13
12
(b)
11 3
12
3
6
1
12
(d)
3 4
18
(e)
13. 14.
5
11
2
1
1
2
1
0 0
0 0 0 0
1
2
1
0 0 0
1
0
0 0
1
2
0
0
3 9 5
0
0
0 0 1
0 0 0
1
2
393
3
12 0
4 5
0
1
Also, diagonalize each of these forms over Q. Carry out the details of the proof of Lemma 2.41. Analogous to the definition of even, we could make the following definition:
Let R be a PID and p a prime not dividing 2 (e.g., R = Z and p an odd
prime). A form 0 is pary if O(x, x) E pR for every x E M. Show that if 0 is pary, then O(x, y) E pR for every x, y E M. 15. Prove Proposition 3.4. 16. Prove Lemma 3.6. 17. Prove Proposition 3.11. 18. Prove Lemma 3.19. 19. Prove Lemma 3.20. 20.
Let f E AUtR(M) where R = R or C and where rank(M) < oo. If fk = Im
21.
for some k, prove that there is a nonsingular form 0 on M with f E Isom(O). Diagonalize the following forms over the indicated fields: (a)
13
4
6
4 6
7
8
8
2
1
(b)
4
1
4
over F2, F3, F5, and Q 1
2
10
10
4
over F2, F3, F5, and Q
22. A symmetric matrix A = (a;2 J E M (R) is called diagonally dominant if laii l
a1+ ?
i#l for 1 < i < n. If the inequality is strict, then A is called strictly diagonally dominant. Let 0 be the bilinear form on R" whose matrix (in the standard basis) is A.
(a) If A is diagonally dominant, show that 0 is positive semidefinite, i.e., O(x, x) >0 for all x E R". (b) If A is strictly diagonally dominant, show that 0 is positive definite. 23. Let 0 be an arbitrary positive (or negative) semidefinite form. Show that is nondegenerate if and only if it is positive (negative) definite. 24. Let R be a ring with a (possibly trivial) conjugation. Show that
{P E GL(n, R) : Pt = (F)_'} is a subgroup of GL n, R). If R = R, with trivial conjugation, this group is called the orthogonal group and denoted O(n), and if R = C, with complex conjugation, it is called the unitary group and denoted U(n).
394 25.
Chapter 6. Bilinear and Quadratic Forms
Let A E M2(C) be the matrix of a Hermitian form, so
A= ab
b
c
with a, c E R. Find an explicit matrix P with P` = (P)' such that P`AP is diagonal. 26.
Let F be an arbitrary subfield of C and 0 a Hermitian form on an Fmodule
M of finite rank. Show how to modify the GramSchmidt procedure to produce an orthogonal (but not in general orthonormal) basis of M, i.e., a
basis B={vl,...,vn} in which O(v;,v,)=0ifi54 j. 27.
Let R be a commutative ring and let A E MM(R) be a skewsymmetric matrix. If P E Mn(R) is any matrix, then show that PAP is also skewsymmetric and
Pf(P`AP) = det(P)Pf(A). 28.
Let A E be a Hermitian matrix. Show that A is positive definite if and only if A1 is positive definite. More generally, show that the signature of A' is the signature of A. (We say that A is positive definite if the associated Hermitian form 0(x, x) = x1Ax is positive definite.)
29.
If V is a real vector space, then a nonempty subset C C V is a cone if a, bEC=a+bECand C. Prove that the set of positive definite Hermitian matrices in Mn (C) is a cone.
30.
If A is a real symmetric matrix prove that there is a E R such that A+aIn is positive definite.
If A E M,n,n(C), show that A'A and AA' are positive semidefinite Hermitian matrices. 32. Let A, B E Mn(C) be Hermitian and assume that B is positive definite. Prove that the two Hermitian forms determined by A and B can be simultaneously diagonalized. That is, prove that there is a nonsingular matrix P 31.
such that P`BP = In and PAP = diag(A1, ... ,,\n). (Hint: B determines an inner product on C".)
Chapter 7
Topics in Module Theory
This chapter will be concerned with collecting a number of results and constructions concerning modules over (primarily) noncommutative rings that will be needed to study group representation theory in Chapter 8.
7.1 Simple and Semisimple Rings and Modules In this section we investigate the question of decomposing modules into "simpler" modules.
(1.1) Definition. If R is a ring (not necessarily commutative) and M 54 (0) is a nonzero Rmodule, then we say that M is a simple or irreducible Rmodule if (0) and M are the only submodules of M.
(1.2) Proposition. If an Rmodule M is simple, then it is cyclic.
Proof. Let x be a nonzero element of M and let N = (x) be the cyclic submodule generated by x. Since M is simple and N # (0), it follows that
M=N.
0
(1.3) Proposition. If R is a ring, then a cyclic Rmodule M = (m) is simple if and only if Ann(m) is a maximal left ideal. Proof. By Proposition 3.2.15, M 91 R/ Ann(m), so the correspondence theorem (Theorem 3.2.7) shows that M has no submodules other than M and (0) if and only if R has no submodules (i.e., left ideals) containing Ann(m) other than R and Ann(m). But this is precisely the condition for Ann(m) to be a maximal left ideal. 0 (1.4) Examples.
(1) An abelian group A is a simple Zmodule if and only if A is a cyclic group of prime order.
Chapter 7. Topics in Module Theory
396
(2) The hypothesis in Proposition 1.3 that M be cyclic is necessary. The Zmodule A = Z2 has annihilator 2Z but the module A is not simple. (3) Consider the vector space F2 (where F is any field) as an F(x]module via the linear transformation T(ul, u2) = (u2i 0). Then F2 is a cyclic F[X]module, but it is not a simple F[X]module. Indeed,
but N = {(u, 0) : u E F} is an F(X]submodule of F2. Thus the converse of Proposition 1.2 is not true. (4) Let V = R2 and consider the linear transformation T : V V defined by T(u, v) (v, u). Then the R[X]module VT is simple. To see
this let w = (ul, v1) # 0 E V and let N be the R[X]submodule of VT generated by w. Then w E N and Xw = T(w) = (vi, ul) E N. Since any (x, y) E V can be written as (x, y) = aw + QXw where a = (xul + yvl)/(ui + vi) and Q = (yul  xvl)/(ui + vi), it follows that N = VT and hence VT is simple. (5) Now let W = C2 and consider the linear transformation T : W + W defined by T(u, v) = (v, u). Note that T is defined by the same formula used in Example 1.4 (4). However, in this case the C(X]module
WT is not simple. Indeed, the Csubspace C (i, 1) is a Tinvariant subspace of W, and hence, it is a C[X]submodule of WT different from W and from (0). The following lemma is very easy, but it turns out to be extremely useful:
(1.5) Proposition. (Schur's lemma) (1) Let M be a simple Rmodule. Then the ring EndR(M) is a division
ring. (2)
If M and N are simple Rmodules, then HomR(M, N) $ (0) if and only if M and N are isomorphic.
Proof. (1) Let f 96 0 E EndR(M). Then Im(f) is a nonzero submodule of M and Ker(f) is a submodule of M different from M. Since M is simple, it follows that Im(f) = M and Ker(f) = (0), so f is an Rmodule isomorphism and hence is invertible as an element of the ring EndR(M). (2) The same argument as in (1) shows that any nonzero homomor
phism f : M N is an isomorphism. We have a second concept of decomposition of modules into simpler pieces, with simple modules again being the building blocks.
(1.6) Definition. If R is a ring (not necessarily commutative), then an Rmodule M is said to be indecomposable if it has no proper nontrivial com
7.1 Simple and Semisimple Rings and Modules
397
plemented submodule M1, i.e., if M = M1 ® M2 implies that M1 = (0) or M1 = M.
If M is a simple Rmodule, then M is also indecomposable, but the converse is false. For example, Z is an indecomposable Zmodule, but Z is not a simple Zmodule; note that Z contains the proper submodule 2Z. One of the major classes of modules we wish to study is the following: (1.7) Definition. An Rmodule M is said to be semisimple if it is a direct sum of simple Rmodules.
The idea of semisimple modules is to study modules by decomposing them into a direct sum of simple submodules. In our study of groups there was also another way to construct groups from simpler groups, namely, the extension of one group by another, of which a special case was the semidirect
product. Recall from Definition 1.6.6 that a group G is an extension of a group N by a group H if there is an exact sequence of groups 1
If this exact sequence is a split exact sequence, then G is a semidirect product of N and H. In the case of abelian groups, semidirect and direct products coincide, but extension of N by H is still a distinct concept. If G is an abelian group and N is a subgroup, then the exact sequence
(0)NpGiH,(0) is completely determined by the chain of subgroups (0) C N C G. By allowing longer chains of subgroups, we can consider a group as obtained by multiple extensions. We will consider this concept within the class of Rmodules.
(1.8) Definition. (1) If R is a ring (not necessarily commutative) and M is an Rmodule, then a chain of submodules of M is a sequence {M;} o of submodules
of M such that (1.1)
(0)=M0 5M15M2
The length of the chain is n. (2)
We say that a chain (Njo is a refinement of the chain {M;}
o
if each M, is equal to N3 for some j. Refinement of chains defines a partial order on the set C of all chains of submodules of M. (3) A maximal element of C (if it exists) is called a composition series of M.
398
Chapter 7. Topics in Module Theory
(1.9) Remarks.
(1) Note that the chain (1.1) is a composition series if and only if each of the modules M;/M;_1 (1 < i < n) is a simple module. (2) Our primary interest will be in decomposing a module as a direct sum of simple modules. Note that if M = ® 1 M; where M; is a simple Rmodule, then M has a composition series n
(0) 511115Ml®M25...5®M:=M. ;=1
On the other hand, if M = ®°_M;, then M does not have a composition series. In a moment (Example 1.10 (2)) we shall see an example of a module that is not semisimple but does have a composition se
ries. Thus, while these two propertiessemisimplicity and having a composition seriesare related, neither implies the other. However, our main interest in composition series is as a tool in deriving results about semisimple modules.
(1.10) Examples. (1)
Let D be a division ring and let M be a Dmodule with a basis
{xl,...,xm}. LetMo=(0)andfor1G(a)I for all a E A, where ISI denotes the cardinality of the set S. We will now present some alternative characterizations of semisimple modules. The following notation, which will be used only in this section, will be convenient for this purpose. Let {Mi}iEI be a set of submodules of a module M. Then let
MI=I: M. iEI
be the sum of the submodules {Mil$E,. (1.21) Lemma. Let M be an Rmodule that is a sum of simple submodules {Mi}iEI, and let N be an arbitrary submodule of M. Then there is a subset
J C I such that
404
Chapter 7. Topics in Module Theory
M'=N®(®Mi). iEJ
Proof. The proof is an application of Zorn's lemma. Let
S={PcI:Mp2, ®MiandMpnN=(0)}. iEP
Partially order S by inclusion and let C = {PQ}QEA be an arbitrary chain
in S. If P = UQEAPQ, we claim that P E S. Suppose that P 0 S. Since it is clear that Mp n N = (0), we must have that Mp ®iEPMI. Then Theorem 3.3.2 shows that there is some po E P, such that Mpo nMp' 0 (0), where P' = P\ {po}. Suppose that 0 96 x E Mpo nMp'. Then we may write (1.8)
x=xp,
where xp, 0 0 E Mp, for {pi, ... ,pk} C P. Since C is a chain, there is an index ct E A such that {po, pi, ... ,pie} C PQ. Equation (1.8) shows that Mpo 9t EIEP, MI, which contradicts the fact that PQ E S. Therefore, we must have P E S, and Zorn's lemma applies to conclude that S has a maximal element J. Claim.
N®((DiEJM1)
If this were not true, then there would be an index io E I such that Mio 0 N + Mj. This implies that Mio ¢ N and Mio ¢ M. Since Mio n N and M,, n Mj are proper submodules of Mio, it follows that M,o n N = (0) and MionMJ = (0) because Mio is a simple Rmodule. Therefore, {io}UJ E 0 S, contradicting the maximality of J. Hence, the claim is proved.
(1.22) Corollary. If an Rmodule M is a sum of simple submodules, then M is semisimple. Proof. Take N = (0) in Theorem 1.21.
0
(1.23) Theorem. If M is an Rmodule, then the following are equivalent:
(1) M is a semisimple module. (2) Every submodule of M is complemented. (3) Every submodule of M is a sum of simple Rmodules.
(1) is immediate Proof. (1) . (2) follows from Lemma 1.21, and (3) from Corollary 1.22. It remains to prove (2) = (3). Let Ml be a submodule of M. First we observe that every submodule of Ml is complemented in Ml. To see this, suppose that N is any submodule of MI. Then N is complemented in M, so there is a submodule N' of M such
7.1 Simple and Semisimple Rings and Modules
405
that N®N' M. But then N+(N'nM1) = M1 so that N® (N'nM1) 2e M1, and hence N is complemented in M1.
Next we claim that every nonzero submodule M2 of M contains a nonzero simple submodule. Let m E M2, m A 0. Then Rm C M2 and, furthermore, R/ Ann(m) S5 Rm where Ann(m) = {a E R : am = 01 is a left ideal of R. A simple Zorn's lemma argument (see the proof of Theorem 2.2.16) shows that there is a maximal left ideal I of R containing Ann(m). Then Im is a maximal submodule of Rm by the correspondence theorem. By the previous paragraph, Im is a complemented submodule of Rm, so
there is a submodule N of Rm with N ® Im L Rm, and since Im is a maximal submodule of Rm, it follows that the submodule N is simple. Therefore, we have produced a simple submodule of M2.
Now consider an arbitrary submodule N of M, and let N1 C N be the sum of all the simple submodules of N. We claim that N1 = N. N1 is complemented in N, so we may write N S5 N1 ® N2. If N2 96 (0) then N2 has a nonzero simple submodule N', and since N' C N, it follows that N' C N1. But N1 n N2 = (0). This contradiction shows that N2 = (0), i.e., N = N1, and the proof is complete. (1.24) Corollary. Sums, submodules, and quotient modules of semisimple modules are semisimple. Proof. Sums: This follows immediately from Corollary 1.22. Submodules: Any submodule of a semisimple module satisfies condition (3) of Theorem 1.23. Quotient modules: If M is a semisimple module, N C M is a submod
ule, and Q = M/N, then N has a complement N' in M, i.e., M ^_' N ®N'. But then Q N', so Q is isomorphic to a submodule of M, and hence, is semisimple.
(1.25) Corollary. Let M be a semisimple Rmodule and let N C M be a submodule. Then N is irreducible (simple) if and only if N is indecomposable.
Proof. Since every irreducible module is indecomposable, we need to show
that if N is not irreducible, then N is not indecomposable. Let N1 be a nontrivial proper submodule of N. Then N is semisimple by Corollary 1.24, so N1 has a complement by Theorem 1.23, and N is not indecomposable. 0
(1.26) Remark. The fact that every submodule of a semisimple Rmodule M is complemented is equivalent (by Theorem 3.3.9) to the statement that whenever M is a semisimple Rmodule, every short exact sequence
O + N + M . K p 0 of Rmodules splits.
406
Chapter 7. Topics in Module Theory
(1.27) Definition. A ring R is called semisimple if R is semisimple as a left Rmodule.
Remark. The proper terminology should be "left semisimple," with an analogous definition of "right semisimple," but we shall see below that the two notions coincide.
(1.28) Theorem. The following are equivalent for a ring R:
(1) R is a semisimple ring. (2) Every Rmodule is semisimple. (3) Every Rmodule is projective.
Proof. (1) . (2). Let M be an Rmodule. By Proposition 3.4.14, M has a free presentation
0+K
FyM40
so that M is a quotient of the free Rmodule F. Since F is a direct sum of copies of R and R is assumed to be semisimple, it follows that F is semisimple, and hence M is also (Corollary 1.24). (2) (3). Assume that every Rmodule is semisimple, and let P be an arbitrary Rmodule. Suppose that (1.9)
is a short exact sequence. Since M is an Rmodule, our assumption is that it is semisimple and then Remark 1.26 implies that sequence (1.9) is split exact. Since (1.9) is an arbitrary short exact sequence with P on the right, it follows from Theorem 3.5.1 that P is projective. (1). Let M be an arbitrary submodule of R (i.e., an arbitrary (3) left ideal). Then we have a short exact sequence
Since all Rmodules are assumed projective, we have that RIM is projective, and hence (by Theorem 3.5.1) this sequence splits. Therefore, M ® N for some submodule N C R, which is isomorphic (as an R Rmodule) to RIM. Then by Theorem 1.23, R is semisimple.
0
(1.29) Corollary. Let R be a semisimple ring and let M be an Rmodule. Then M is irreducible (simple) if and only if M is indecomposable. Proof.
11
(1.30) Theorem. Let R be a semisimple ring. Then every simple Rmodule is isomorphic to a submodule of R.
Proof. Let N be a simple Rmodule, and let R = ®IE!Mi be a simple factorization of the semisimple Rmodule R. We must show that at least
7.1 Simple and Semisimple Rings and Modules
407
one of the simple Rmodules M; is isomorphic to N. If this is not the case, then
HomR ((DM,, N) = II HomR(Mi, N) = (0)
HOMR(R, N)
iEI
iEI
where the last equality is because HomR(Mi, N) = (0) if Mi is not isomorphic to N (Schur's lemma). But HomR(R, N) N 0 (0), and this contradiction shows that we must have N isomorphic to one of the simple
0
submodules Mi of R. (1.31) Corollary. Let R be a semisimple ring.
(1) There are only finitely many isomorphism classes of simple Rmodules. (2) If {Ma}QEA is the set of isomorphism classes of simple Rmodules and
R = ®(I'aMQ), oEA
then each ra is finite. Proof. Since R is semisimple, we may write
R= ®Np OEB
where each No is simple. We will show that B is finite, and then both finiteness statements in the corollary are immediate from Theorem 1.30. Consider the identity element 1 E R. By the definition of direct sum, we have
1= 1: rpnp OE B
for some elements rp E R, np E N0, with all but finitely many rp equal to zero. Of course, each Np is a left Rsubmodule of R, i.e., a left ideal.
Now suppose that B is infinite. Then there is a go E B for which r00 = 0. Let n be any nonzero element of Npo. Then
n = n 1 = n(E rpnp) _
F,
(nrp)np,
pEB\{po}
pEB so
nE ® Np. pEB\{po}
Thus,
n E No. n( @ No) _ {0}, pEB\{po}
Chapter 7. Topics in Module Theory
408
by the definition of direct sum again, which is a contradiction. Hence, B is finite.
We now come to the basic structure theorem for semisimple rings.
(1.32) Theorem. (Wedderburn) Let R be a semisimple ring. Then there is a finite collection of integers n1, ... , nk, and division rings D1, ... , Dk such that k
R
(DEndD,(D°`). i=1
Proof. By Corollary 1.31, we may write k
R®niMi i=1
where {M,}k 1 are the distinct simple Rmodules and nl, ... , nk are positive integers. Then R is antiisomorphic to R°P, and R°P = EndR(R) k
k
(
HomR (® niMi, ® niMil/ i=1
i=1
k
®HomR(niMM, niMi) i=1 k
^_' ®EndR(niM,), i=1
by Schur's lemma. Also, by Schur's lemma, EndR(MM) is a division ring,
which we denote by Ei, for each i = 1, ..., k. Then it is easy to check (compare the proof of Theorem 1.18) that EndR(n1Mi)
EndE.(E;')
Setting Di = E,'P, the proof is completed by observing that EndE, (E; ") is antiisomorphic to EndD, (Dr ). Remark. Note that by Corollary 4.3.9, EndD(D") is isomorphic to M,, (DP). Thus, Wedderburn's theorem is often stated as, Every semisimple ring is isomorphic to a finite direct sum of matrix rings over division rings.
(1.33) Lemma. Let D be a division ring and n a positive integer. Then R = EndD(D") is semisimple as a left Rmodule and also as a right Rmodule. Fhrthermore, R is semisimple as a left Dmodule and as a right Dmodule.
7.1 Simple and Semisimple Rings and Modules
409
Proof. Write D' = Dl ®D2 ® ® D,, where Di = D. Let
Mi = If E EndD(D") : Ker(f) 2 (DD, k#i
Nj = {f E EndD(D") : Im(f) C Dj}, and let
Pij=MinN.. Note that Pij =' D°p. Then
EndD(D") `= Ml ®... ®M,, as a left Rmodule, and EndD(D")2?5
as a right Rmodule. We leave it to the reader to check that each Mi (resp., Nj) is a simple left (resp., right) Rmodule. Also,
EndD(D") = ® Pij as a right (resp., left) Dmodule, and each Pij is certainly simple (on either side).
(1.34) Corollary. A ring R is semisimple as a left Rmodule if and only if it is semisimple as a right Rmodule. Proof. This follows immediately from Theorem 1.32 and Lemma 1.33.
Observe that R is a simple left Rmodule (resp., right Rmodule) if and only if R has no nontrivial proper left (resp., right) ideals, which is the case if and only if R is a division algebra. Thus, to define simplicity of R in this way would bring nothing new. Instead we make the following definition:
(1.35) Definition. A ring R with identity is simple if it has no nontrivial proper (twosided) ideals.
Remark. In the language of the next section, this definition becomes "A ring R with identity is simple if it is simple as an (R, R)bimodule." (1.36) Corollary. Let D be a division ring and n a positive integer. Then EndD(D") is a simple ring that is semisimple as a left Conversely, if R is a simple ring that is semisimple as a left Rmodule, or, equivalently, as a right Rmodule, then R
EndD(D")
for some division ring D and positive integer n.
410
Chapter 7. Topics in Module Theory
Proof. We leave it to the reader to check that Endo(D's) is simple (compare Theorem 2.2.26 and Corollary 2.2.27), and then the first part of the corollary follows from Lemma 1.33. Conversely, if R is semisimple we have the decomposition given by Wedderburn's theorem (Theorem 1.32), and then the condition of simplicity forces k = 1.
Our main interest in semisimple rings and modules is in connection with our investigation of group representation theory, but it is also of interest to reconsider modules over a PID from this point of view. Thus let R be a PID. We wish to give a criterion for Rmodules to be semisimple. The following easy lemma is left as an exercise. (1.37) Lemma. Let R be an integral domain. Then R is a semisimple ring if and only if R is a field. If R is a field, R is simple. Proof. Exercise.
From this lemma and Theorem 1.28, we see that if R is a field, then every Rmodule (i.e., vector space) is semisimple and there is nothing more to say. For the remainder of this section, we will assume that R is a PID that is not a field. Let M be a finitely generated Rmodule. Then by Corollary 3.6.9, we
have that M F ® M where F is free (of finite rank) and Mr is the torsion submodule of M. If F 76 (0) then Lemma 1.37 shows that M is not semisimple. It remains to consider the case where M = M i.e., where M is a finitely generated torsion module. Recall from Theorem 3.7.13 that each such M is a direct sum of primary cyclic Rmodules. (1.38) Proposition. Let M be a primary cyclic Rmodule (where R is a PID is not a field) and assume that Ann(M) = (pe) where p E R is a prime. If e = 1 then M is simple. If e > 1, then M is not semisimple.
Proof. First suppose that e = 1, so that M°5 R/(p). Then M is a simple Rmodule because (p) is a prime ideal in the PID R, and hence, it is a maximal ideal.
Next suppose that e > 1. Then (0) A p`1 M 5 M,
and pe! M is a proper submodule of M, which is not complemented; hence, M is not semisimple by Theorem 1.23 (2).
(1.39) Theorem. Let M be a finitely generated torsion Rmodule (where R is a PID that is not a field). Then M is semisimple if and only if me(M)
7.1 Simple and Semisimple Rings and Modules
411
(see Definition 3.7.8) is a product of distinct prime factors. M is a simple Rmodule if and only if
me(M) = co(M) = (p) where p E R is a prime.
Proof. First suppose that M is cyclic, and me(M) = (pi' ... pr). Then the primary decomposition of M is given by MSM (R/(pi'))ED ...(D (Rl
k'')),
and M is semisimple if and only if each of the summands is semisimple, which by Proposition 1.38, is true if and only if ei=e2=...=ek=1.
Now let M be general. Then by Theorem 3.7.1, there is a cyclic decomposition
Rw such that Ann(wi) = (si) and si si+l for 1 < i < n  1. Then M is [
semisimple if and only if each of the cyclic submodules Rwi is semisimple, which occurs (by the previous paragraph) if and only if si is a product of distinct prime factors. Since si I si+i, this occurs if and only if s = me(M) is a product of distinct prime factors. The second assertion is then easy to verify.
0
(1.40) Remark. In the two special cases of finite abelian groups and linear transformations that we considered in some detail in Chapters 3 and 4, Theorem 1.39 takes the following form: (1) A finite abelian group is semisimple if and only if it is the direct product of cyclic groups of prime order, and it is simple if and only if it is cyclic of prime order.
(2) Let V be a finitedimensional vector space over a field F and let T : V + V be a linear transformation. Then VT is a semisimple F[XJmodule if and only if the minimal polynomial mT(X) of T is a product of distinct irreducible factors and is simple if and only if its characteristic polynomial cT(X) is equal to its minimal polynomial mT(X), this polynomial being irreducible (see Lemma 4.4.11.) If F is algebraically closed (so that the only irreducible polynomials are linear ones) then VT is semisimple if and only if T is diagonalizable and simple if and only if V is onedimensional (see Corollary 4.4.32).
412
Chapter 7. Topics in Module Theory
7.2 Multilinear Algebra We have three goals in this section: to introduce the notion of a bimodule, to further our investigation of "Hom," and to introduce and investigate tensor
products. The level of generality of the material presented in this section is dictated by the applications to the theory of group representations. For this reason, most of the results will be concerned with modules over rings that are not commutative; frequently there will be more than one module structure on the same abelian group, and many of the results are concerned with the interaction of these various module structures. We start with the concept of bimodule.
(2.1) Definition. Let R and S be rings. An abelian group M is an (R, S)bimodule if M is both a left Rmodule and a right Smodule, and the compatibility condition
r(ms) = (rm)s
(2.1)
is satisfied for every r E R, m E M, and s E S. (2.2) Examples. (1)
(2)
Every left Rmodule is an (R, Z)bimodule, and every right Smodule is a (Z, S)bimodule.
If R is a commutative ring, then every left or right Rmodule is an (R, R)bimodule in a natural way. Indeed, if M is a left Rmodule, then according to Remark 3.1.2 (1), M is also a right Rmodule by means of the operation r,ir = rm. Then Equation (2.1) is
r(ms) = r(sm) = (rs)m = (sr)m = s(rm) = (rm)s. (3) If T is a ring and R and S are subrings of T (possibly with R = S = T),
then T is an (R, S)bimodule. Note that Equation (2.1) is simply the associative law in T. (4) If M and N are left Rmodules, then the abelian group HomR(M, N)
has the structure of an (EndR(N), EndR(M))bimodule, as follows. If
f E HomR(M, N), ¢ E EndR(M), and V) E EndR(N), then define f ¢ = f o 0 and f = o f . These definitions provide a left EndR(N)module and a right EndR(M)module structure on HomR(M, N), and Equation (2.1) follows from the associativity of composition of functions. (5)
Recall that a ring T is an Ralgebra, if T is an Rmodule and the Rmodule structure on T and the ring structure of T are compatible, i.e.,
r(tit2) = (rtl)t2 = ti(rt2) for all r E R and t1, t2 E T. If T happens to be an (R,S)bimodule, such that r(tlt2) = (rtl)t2 = tl(rt2) and (tlt2)s = tl(t2s) = (tls)t2 for all r E R, s E S, and t1, t2 E T, then we
7.2 Multilinear Algebra
413
say that T is an (R, S)blalgebra. For example, if R and S are subrings of a commutative ring T, then T is an (R, S)bialgebra. Suppose that M is an (R, S)bimodule and N C M is a subgroup of the additive abelian group of M. Then N is said to be an (R, S)bisubmodule of M if N is both a left Rsubmodule and a right Ssubmodule of M. If Ml and M2 are (R, S)bimodules, then a function f : Ml M2 is an (R, S)bimodule homomorphism if it is both a left Rmodule homomorphism and a right Smodule homomorphism. The set of (R, S)bimodule homomorphisms will be denoted Hom(R,S)(M1, M2). Since bimodule homomorphisms can be added, this has the structure of an abelian group,
but, a priori, nothing more. If f : Ml + M2 is an (R, S)bimodule homomorphism, then it is a simple exercise to check that Ker(f) C Ml and Im(f) C M2 are (R, S)bisubmodules. Furthermore, if N C M is an (R, S)bisubmodule, then the quotient abelian group is easily seen to have the structure of an (R, S)bimodule. We leave it as an exercise for the reader to formulate and verify the noether isomorphism theorems (see Theorems 3.2.3 to 3.2.6) in the context of (R, S)bimodules. It is worth pointing out that if M is an (R, S)bimodule, then there are three distinct concepts of submodule of M, namely, Rsubmodule, Ssubmodule, and (R, S)bisubmodule. Thus, if X C M, then one has three
concepts of submodule of M generated by the set X. To appreciate the difference, suppose that X = {x} consists of a single element x E M. Then the Rsubmodule generated by X is the set
Rx = {rx : r E R},
(2.2)
the Ssubmodule generated by X is the set XS = {X8: s E S},
(2.3)
while the (R, S)bisubmodule generated by X is the set (2.4)
RxS={>r;xsj:nENandr1ER,siESfor1[i o 0, so we
(2.13) Remarks. (1) If M is a right Rmodule and N is a left Rmodule, then M OR N is
an abelian group. (2) If M and N are both (R, R)bimodules, then M OR N is an (R, R)
bimodule. A particular (important) case of this occurs when R is a commutative ring. In this case every left Rmodule is automatically a right Rmodule, and viceversa. Thus, over a commutative ring R, it is meaningful to speak of the tensor product of Rmodules, without explicit attention to the subtleties of bimodule structures. (3) Suppose that M is a left Rmodule and S is a ring that contains R as a subring. Then we can form the tensor product S®R M which has the structure of an (S, Z)bimodule, i.e, S OR M is a left Smodule. This construction is called change of rings and it is useful when one would like to be able to multiply elements of M by scalars from a bigger ring. For example, if V is any vector space over R, then C ®R V is a vector space over the complex numbers. This construction has been implicitly used in the proof of Theorem 4.6.23. (4) If R is a commutative ring, M a free Rmodule, and 0 a bilinear form on M, then 0: M x M  R is certainly middle linear, and so 0 induces an Rmodule homomorphism
4:M®RM.R.
7.2 Multilinear Algebra
421
(2.14) Corollary. (1) Let M and M' be (R, S)bimodules, let N and N' be (S,T)bimodules,
and suppose that f : M + M' and g : N + N' are bimodule homomorphisms. Then there is a unique (R,T)bimodule homomorphism (2.20)
f ®g=f Os g: M®sN +M'®sN'
satisfying (f ®g) (m (9 n) = f (m) ®g(n) for all m E M, n E N. (2) If M" is another (R, S)bimodule, N" is an (S, T)bimodule, and f" :
M' . M", g" : N' . N" are bimodule homomorphisme, then letting
f®g:M®N+M'®N' and f'®g':M'®N'WON" be defined as in part (1), we have (f'®9)(f®g)=(f'f)0(9g):M®N+M"®N".
Proof. (1) Let F be the free abelian group on M x N used in the definition of M®s N' be the unique Zmodule homomorphism M ®s N, and let h : F such that h(m, n) = f (m) Os g(n). Since f and g are bimodule homomorphisms, it is easy to check that h is an Smiddle linear map, so by Theorem
2.12, there is a unique bimodule homomorphism h : M 0 N + M' ® N' such that h = h o 0 where 0 : M x N + M 0 N is the canonical map. Let f ®g = h. Then (f 0 g)(m (9 n) = h(m (9 n) = h o 0(m, n) = h(m, n) = f (m) 0 g(n) as claimed. (2) is a routine calculation, which is left as an exercise.
0
We will now consider some of the standard canonical isomorphisms relating various tensor product modules. The verifications are, for the most part, straightforward applications of Theorem 2.12. A few representative calculations will be presented; the others are left as exercises.
(2.15) Proposition. Let M be an (R, S)bimodule. Then there are (R, S)bimodule isomorphisms
R®RM5M and
M®sS:L, M.
Proof. We check the first isomorphism; the second is similar. Let f : R x M ' M be defined by f (r, m) = rm. It is easy to check that f is an Rmiddle linear map, and thus Theorem 2.12 gives an (R, S)bimodule
homomorphism f : R OR M . M such that A r 0 m) = rm. Define g : M + R OR M by g(m) = 1 ® m. Then g is an (R, S)bimodule homomorphism, and it is immediate that f and g are inverses of each other. 0
422
Chapter 7. Topics in Module Theory
(2.16) Proposition. Let R be a commutative ring and let M and N be Rmodules. Then
M®IIN=N®RM. Proof. The isomorphism is given (via an application of Theorem 2.12) by
0 (2.17) Proposition. Let M be an (R,S)bimodule, N an (S, T)bimodule, and P a (T, U)bimodule. Then there is an isomorphism of (R, U)bimodules
(M ®s N) OT
MOs (N®TP).
Proof. Fix an element p E P and define a function
fp:MXNIM®s(N®TP) by
fp(m, n) = m ®s (n ®T p)
fp is easily checked to be Smiddle linear, so Theorem 2.12 applies to give an (R, T)bimodule homomorphism fp : M ®s N a M ®s (N ®T P). Then we have a map f : (M ®s N) x P + lvi ®s (N ®T P) defined by
f ((m ®n), p) = fp(m ®n) = m ®(n (& p)
But f is Tmiddle linear, and hence there is a map of (R, U)bimodules
f: (M os N) OT P 
M®s(N®TP)
satisfying f ((m®n)®p) = m®(n®p). Similarly, there is an (R, U)bimodule homomorphism
g:M®s(NOT P) (M®sN)®TP satisfying gg(m (9 (n ®p)) = (m (9 n) ®p. Clearly, §1 (respectively j g) is the identity on elements of the form (m ®n) ®p (respectively, m ®(n (&p)), and since these elements generate the respective tensor products, we conclude
that f and g are isomorphisms.
O
(2.18) Proposition. Let M = ®iEIMi be a direct sum of (R, S)bimodules, and let N = ®)EJNj be a direct sum of (S,T)bimodules. Then there is an isomorphism
M®sN®®(MM®sN3) iEI jEJ of (R, T) bimodules.
Proof. Exercise.
O
7.2 Multilinear Algebra
423
(2.19) Remark. When one is taking Horn and tensor product of various bimodules, it can be somewhat difficult to keep track of precisely what type of module structure is present on the given Hom or tensor product. The following is a useful mnemonic device for keeping track of the various module structures when forming Hom and tensor products. We shall write RMS to indicate that M is an (R, S)birnodule. When we form the tensor product of an (R, S)birnodule and an (S, T)birnodule, then the resulting module has the structure of an (R, T)bimodule (Definition 2.9). This can be indicated mnemonically by (2.21)
RMS ®S SNT = RPT
Note that the two subscripts "S" on the bimodules appear adjacent to the subscript "S" on the tensor product sign, and after forming the tensor product they all disappear leaving the outside subscripts to denote the bimodule type of the answer (= tensor product).
A similar situation holds for Horn, but with one important difference. Recall from Proposition 2.4 that if M is an (R, S)bimodule and N is an (R, T)bimodule, then HornR(M, N) has the structure of an (S,T)bimodule. (Recall that HomR(M, N) denotes the left Rmodule homomorphisms.) In order to create a simple mnemonic device similar to that of Equation (2.21), we make the following definition. If M and N are left Rmodules, then we will write M fiR N for HomR(M, N). Using mR in place of OR, we obtain the same convention about matching subscripts disappearing, leaving the outer subscripts to give the bimodule type, provided that the order of the subscripts of the module on the left of the OR sign are reversed. Thus, Proposition 2.4 is encoded in this context as the statement
RMS and
RNT . SMR 'I'R RNT = SPT A similar convention holds for homomorphisms of right Tmodules. This is illustrated by
Hom_T(RMT, SNT) = SNT 'I'T TMR = SPR, AA,,
the result being an (S, R)birnodule (see Proposition 2.5). Note that we must reverse the subscripts on M and interchange the position of M and N. We shall now investigate the connection between Horn and tensor product. This relationship will allow us to deduce the effect of tensor products on exact sequences, using the known results for Hom (Theorems 2.7 and 2.8 in the current section, which are generalizations of Theorems 3.3.10 and 3.3.12).
(2.20) Theorem. (Adjoint associativity of Hom and tensor product) Let Ml and M2 be (S, R)bimodules, N a (T, S)bimodule, and P a (T, U)bimodule. If 0 : M2 , Ml is an (S, R)bimodule homomorphism, then
Chapter 7. Topics in Module Theory
424
there are (R, U)bimodule isomorphisms
Homs(Mi, HomT(N, P))
HomT(N (&s Mi, P)
such that the following diagram commutes:
Homs(Mi, HomT(N, P))
f
Homs(M2, HomT(N, P))
(2.22)
1 02
HOMT(N (&s Ml, P) Proof. Define
(IN
O).
____+
HOMT(N OS M2, P)
Homs(Mi, HomT(N, P)) + HomT(N (&s Mi, P) by I{(f)(n (& m) = (f (m)) (n)
where f E Homs(Mi, HorT(N, P)), m E Mi, and n E N. It is easy to check that 4i (f) E HomT (N (&s M, P) and that $ is a homomorphism of (R, U)bimodules. The inverse map is given by ('Ii(g)(m)) (n) = g('m ® n)
where g E HomT(N ®s M, P), m E M, and n E N. To check the commutativity of the diagram, suppose that f E Homs(M, HomT(N, P), n E N, and M2 E M2. Then
((4'2oP*)(f))(n(9 m2) _ (42(f o,')) (n(9 m2) _ ((f oV)(m2)) (n) = f (IG(m2))(n)
_ ('1(f )) (n ®?P(ms))
_ (4(f)) ((In (9 ',)(n ® m2)) _ (1n (9 1,)' (`Oi (f )) (n ® m2)
_ ((1N (9'G)* o61(f)) (n(9 m2).
Thus, 02
(IN
o 4i1 and diagram (2.22) is commutative.
0
There is an analogous result concerning homomorphisms of right modules. In general we shall not state results explicitly for right modules; they can usually be obtained by obvious modifications of the left module results. However, the present result is somewhat complicated, so it will be stated precisely.
(2.21) Theorem. Let M1 and M2 be (R, S)bimodules, N an (S, T)bimodule, and P a (U, T)bimodule. If ip : M2 + M1 is an (R, S) bimodule homomorphism, then there are (U, R)bimodule isomorphisms
,ti : Hom_s(M;, Hom_T(N, P)) i Hom_T(Mi ®s N, P)
7.2 Multilinear Algebra
425
such that the following diagram commutes:
Hom_s(Ml, Hom_T(N, P)) (2.23)
j.
Hom_s(M2, Hom_T(N, P)) t'D2
1
(1NuS 1W
Hom_T(M1 ®s N, P)
Hom_T(M2 (&s N, P)
Proof. The proof is the same as that of Theorem 2.20.
Remark. Note that Theorems 2.20 and 2.21 are already important results
in case M1=M2=Mandt=1M. As a simple application of adjoint associativity, there is the following result. (2.22) Corollary. Let M be an (R, S) bimodule, N an (S, T) bimodule, and
let P = M ®s N (which is an (R, T) bimodule). If M is projective as a left Rmodule (resp., as a right Smodule) and N is projective as a left Smodule (resp., as a right Tmodule), then P is projective as a left Rmodule (resp., as a right Tmodule).
Proof. To show that P is projective as a left Rmodule, we must show that, given any surjection f : A B of Rmodules, the induced map
f.: HomR(P, A) > HomR(P, B) is also surjective. By hypothesis, M is projective as a left Rmodule so that
f.: HomR(M, A)
HomR(M, B)
is surjective. Also, N is assumed to be projective as a left Smodule, so the map
(f.).: Homs(N, HomR(M, A)) > Homs(N, HomR(M, A)) is also surjective. But, by Theorem 2.20, if C = A or B, then
Homs(N,HomR(M, C)) = HomR(P, C). It is simple to check that in fact there is a commutative diagram Homs(N, HomR(M, A)) 1
411
HomR(P, A)
and this completes the proof.
(
Homs(N, HomR(M, B))
I
'2
HomR(P, B)
Chapter 7. Topics in Module Theory
426
One of the most important consequences of the adjoint associativity property relating Horn and tensor product is the ability to prove theorems concerning the exactness of sequences of tensor product modules by appealing to the theorems on exactness of Horn sequences, namely, Theorems 2.7 and 2.8. (2.23) Theorem. Let N be a fixed (R, T)bimodule. If
M100.M  M2 +0
(2.24)
is an exact sequence of (S, R)bimodules, then
Ml®RN + M®RNg, 01N M2®RN+0 "Iri
(2.25)
is an exact sequence of (S, T)bimodules, while if (2.24) is an exact sequence of (T, S) bimodules, then
NOT Ml'N®mN®TM'i N®TM2 +0
(2.26)
is an exact sequence of (R, S)bimodules. Proof. We will prove the exactness of sequence (2.26); exactness of sequence (2.25) is similar and it is left as an exercise. According to Theorem 2.7, in order to check the exactness of sequence (2.26), it is sufficient to check that the induced sequence (2.27)
HomR(N ®T M2, P)
0
HomR(N ®T M, P)
+ HomR(N ®T M1, P) is exact for every (R, U)bimodule P. But Theorem 2.20 identifies sequence (2.27) with the following sequence, which is induced from sequence (2.24) by the (T, U)bimodule HomR(N, P): (2.28)
0  HomT(M2, HomR(N, P))  HomT(M, HomR(N, P))  HomT(Ml, HomR(N, P)).
Since (2.24) is assumed to be exact, Theorem 2.7 shows that sequence (2.28)
is exact for any (R, U)bimodule P. Thus sequence (2.27) is exact for all 0 P, and the proof is complete. (2.24) Examples.
(1) Consider the following short exact sequence of Zmodules: (2.29)
0
where 0(i) = mi and i is the canonical projection map. If we take N = Z,,, then exact sequence (2.25) becomes
7.2 Multilinear Algebra
427
Z®Zn +Z®Zn ® Zm®Zn + 0.
(2.30)
By Proposition 2.15, exact sequence (2.30) becomes the exact sequence
Zn  4Zn '1'+ Zm®Zn'0
(2.31)
mi. Thus Zm ® Zn
where
Coker(QS). Now let d = gcd(m, n) is the composite
and write m = m'd, n = n'd. Then the map Zn 01 + Zn
02
+ Zn
where 01(i) = m'i and 02(i) = di. Since gcd(m', n) = 1, it follows that 01 is an isomorphism (Proposition 1.4.11), while IM(02) = dZn. Hence, Coker(q) Zn/dZn Zd, i.e.,
Zm®Zn
Zd.
(2) Suppose that M is any finite abelian group. Then M ®z Q = (0). To see this, consider a typical generator x ®r of M ®z Q, where x E M
and r E Q. Let n = IMI. Then nx = 0 and, according to Equation (2.18),
x ® r = x 0 n(r/n) = xn ® (r/n) = 0 ® (r/n) = 0. Since x E M and r E Q are arbitrary, it follows that every generator
ofM®Q is 0. (3)
Let R be a commutative ring, I C R an ideal, and M any Rmodule. Then (2.32)
(R/I) OR M M/IM.
To see this consider the exact sequence of Rmodules
0+I `+R+R/I+0. Tensor this sequence of Rmodules with M to obtain an exact sequence
I®RM `®+RORM+(R/I) ®RM+0. But according to Proposition 2.15, R®R M = M (via the isomorphism
4i(r (9 m) = rm), and under this identification it is easy to see that Im(c (9 1) = IM. Thus, (R/I) OR M M/IM, as we wished to verify. Example 2.32 (1) shows that even if a sequence
0 + M1 + M ' M2
0
428
Chapter 7. Topics in Module Theory
is short exact, the tensored sequence (2.25) need not be part of a short exact sequence, i.e., the initial map need not be injective. For a simple situation where this occurs, take m = n in Example 2.32 (1). Then exact sequence (2.30) becomes
Zn001 ' Zn  Zn + 0. 1 is the zero map, so it is certainly not an injection. The map This example, plus our experience with Hom, suggests that we consider criteria to ensure that tensoring a short exact sequence with a fixed module produces a short exact sequence. We start with the following result, which is exactly analogous to Theorem 2.8 for Hom. (2.25) Theorem. Let N be a fixed (R, T)bimodule. If
0pM1.M.M2+0
(2.33)
is a split short exact sequence of (S, R) bimodules, then (2.34)
0 Ml®RNO 1 M®RN4
M2®RN+0
is a split short exact sequence of (S, T)bimodules, while if (2.33) is a split short exact sequence of (T, S)bimodules, then (2.35)
0
N ®T M1
14 N ®T M 1 N ®T M2 i 0
is a split short exact sequence of (R, S)bimodules. Proof. We will do sequence (2.34); (2.35) is similar and is left as an exercise.
Let of : M  M1 split 0, and consider the map
a®1:Al OR N'M1OR N. Then
((a®1)(O®1))(m(9 n) = (aO(9 1)(m®n) = (1(9 1)(m(& n) = m®n so that 0®1 is an injection, which is split by a®1. The rest of the exactness 0 is covered by Theorem 2.23.
(2.26) Remark. Theorems 2.7 and 2.23 show that given a short exact sequence, applying Hom or tensor product will give a sequence that is exact on one end or the other, but in general not on both. Thus Horn and tensor product are both called half exact, and more precisely, Hom is called left exact and tensor product is called right exact. We will now investigate some conditions under which the tensor product of a module with a short exact sequence always produces a short exact sequence. It was precisely this type of consideration for Hom that led us to the concept of projective module. In fact, Theorem 3.5.1 (4) shows that if P is a projective Rmodule and
7.2 Multilinear Algebra
429
0 M1 0+M &M2i0 is a short exact sequence of Rmodules, then the sequence
0 + HomR(P, M1) i HomR(P, M) 0.+ HomR(P, M2) + 0 is short exact. According to Theorem 3.3.10, the crucial ingredient needed
is the surjectivity of t/i. and this is what projectivity of P provides. For the case of tensor products, the crucial fact needed to obtain a short exact sequence will be the injectivity of the initial map of the sequence.
(2.27) Proposition. Let N be an (R, T)bimodule that is projective as a left Rmodule. Then for any injection t : M1 ' M of (S, R)bimodules,
t®1:M1®RN +M®RN is an injection of (S, T)bimodules. If N is projective as a right Tmodule and t : M1 + M is an injection of (T, S)bimodules, then
1®t:N®TMi + N®TM is an injection of (R, S)bimodules.
Proof. First suppose that as a left Rmodule N is free with a basis {nj}jEJ.
Then N ®jEJRj where each summand Rj = Rnj is isomorphic to R as a left Rmodule. Then by Proposition 2.18
MI OR N°(D(MlOR Rj)=$Mlj jEJ
jEJ
where each MI j is isomorphic to M1 as a left Smodule, and similarly M®RN ®jEJMj, where each Mj is isomoprhic to M as a left Smodule. Furthermore, the map t ®1 : M1®R N + M OR N is given as a direct sum
®(tj:Mij+Mj) jEJ
where each tj agrees with i under the above identifications. But then, since t is an injection, so is each tj, and hence so is t ®1. Now suppose that N is projective as a left Rmodule. Then there is a left Rmodule N' such that N ® N' = F where F is a free left Rmodule. We have already shown that
t®1:Mi®RF+M®RF is an injection. But using Proposition 2.18 again,
MI OR F= (M1 OR N)®(MI OR N')
430
Chapter 7. Topics in Module Theory
so we may write t(91 = c1® t2 where (in particular) c1 = L(91 : M1®R N + M OR F. Since c ® 1 is an injection, so is c1, as claimed. Thus the proof is
complete in the case that N is projective as a left Rmodule. The proof in case N is projective as a right Tmodule is identical. Note that we have not used the right Tmodule structures in the above proof. This is legitimate, since if a homomorphism is injective as a map of left Smodules, and it is an (S, T)bimodule map, then it is injective as an (S, T)bimodule map.
(2.28) Corollary. Let N be a fixed (R,T)bimodule that is projective as a left Rmodule. If (2.36)
0.M1
is a short exact sequence of (S, R)bimodules, then (2.37)
0+ M1 ORNMORN"0
M2®RN+0
is a short exact sequence of (S, T)bimodules; while if (2.36) is an exact sequence of (T, S)bimodules and N is projective as a right Tmodule, then (2.38)
0+N®TM1' N®TM'  N®TM2+0
is a short exact sequence of (R, S)bimodules. Proof. This follows immediately from Theorem 2.23 and Proposition 2.27. 0
(2.29) Remark. A module satisfying the conclusion of Proposition 2.27 is said to be flat. That is, a left Rmodule N is flat if tensoring with all short exact sequences of right Rmodules produces a short exact sequence, with a similar definition for right Rmodules. Given Theorem 2.23, in order to prove that a left Rmodule N is flat, it is sufficient to prove that for all right Rmodules M and submodules K, the inclusion map e : K + M induces an injective map
t®1:K®RN M®RN.
Thus, what we have proven is that projective modules are flat. In Section 6.1 we discussed duality for free modules over commutative rings. Using the theory developed in the current section, we will extend portions of our discussion of duality to the context of projective (bi)modules.
(2.30) Definition. Let M be an (R, S)bimodule. The dual module of M is the (S, R)bimodule M' defined by HomR(M, R).
7.2 Multilinear Algebra
431
In particular, if M is a left Rmodule, i.e., take S = Z, then the dual module M* is a right Rmodule. The double dual of M is defined to be
M" = Hom_R(M', R). As in Section 6.1, there is a homomorphism rl
: M  M" of (R,S)
bimodules defined by
(rl(v))(w) = w(v) for all v E M, w E M' and if r) is an isomorphism, then we will say that M is reflexive.
If M is an (R, S)bimodule, which is finitely generated and free as a left Rmodule, then given any basis B of M, one may construct a basis of M* (as a right Rmodule) exactly as in Definition 6.1.3 and the proof of Theorem 6.1.7 goes through verbatim to show that finitely generated free Rmodules are reflexive, even when R need not be commutative. Furthermore, the proofs of Theorems 3.5.8 and 6.1.13 go through without difficulty if one keeps track of the types of modules under consideration. We will simply state the following result and leave the details of tracing through the module types as an exercise. (2.31) Proposition. Let M be an (R, S)bimodule, which is finitely generated
and projective as a left Rmodule. Then the dual module M' is finitely generated and projective as a right Rmodule. Furthermore, M is reflexive as an (R, S)bimodule. Proof. Exercise. See the comments above.
0
If M is an (R, S)bimodule and P is an (R, T)bimodule, then define
(:M'xP+HomR(M,P) by
(((w, p)) (m) = w(m)p
for
w E M', p E P, and m E M.
Then ( is Smiddle linear and hence it induces an (S, T)bimodule homomorphism
t;:M'®RPkHomR(M,P)
given by
(2.39)
(((W'& p))(m) = w(m)p
for all w E M*, p E P, and m E M. (2.32) Proposition. Let M be an (R, S)bimodule, which is finitely generated and projective as a left Rmodule, and let P be an arbitrary (R, T) bimodule. Then the map
Chapter 7. Topics in Module Theory
432
( : M' OR P  HomR(M, P) defined by Equation (2.39) is an (S, T) bimodule isomorphism.
Proof. Since S is an (S, T)bimodule homomorphism, it is only necessary to prove that it is bijective. To achieve this first suppose that M is free of finite rank k as a left Rmodule. Let B = {vl, ... , vk} be a basis of M and let {vi, ... ,v;} be the basis of M' dual to B. Note that every element of M' OR P can be written as x = Fk 1 of ®pi for pt, ... , Pk E P. Suppose
that ((x) = 0, i.e., (((x))(m) = 0 for every m E M. But ((x)(vi) = pi so that pi = 0 for 1 < i < k. That is, x = 0 and we conclude that ( is injective. Given any f E HomR(M, P), let k
xf=>2v1 ®f(vi) Then (((xf))(vi) = f (vi) for 1 < i < k, i.e., S(xf) and f agree on a basis of M; hence, ((xf) = f and C is a surjection, and the proof is complete in case M is free of rank k. Now suppose that M is finitely generated and projective, and let N be a left Rmodule such that F = M ® N is finitely generated and free. Then S : F' OR P  HomR(F, P) is a Zmodule isomorphism, and
F'®RP= (M®N)'®RPQ (M'(DN')ORP°` (M* OR P)®(N'®R P) while
HomR(F, P) = HomR(M ® N, P)
HomR(M, P) ® HomR(N, P)
where all isomorphisms are Zmodule isomorphisms. Under these isomorphisms,
CF=CM®(N CM : M' OR P
HomR(M, P)
CN : N' OR P ' HomR(N, P). Since (F is an isomorphism, it follows that CM and (N are isomorphisms as well. In particular, CM is bijective and the proof is complete. 0
(2.33) Corollary. Let M be an (R, S)bimodule, which is finitely generated and projective as a left Rmodule, and let P be an arbitrary (T, R)bimodule. Then
(P®RM)' as (S,T)bimodules.
Proof. From Proposition 2.32, there is an isomorphism
7.2 Multilinear Algebra
M' OR P'
433
HomR(M, P') = HomR(M, HomR(P, R)) (by adjoint associativity)
LY HomR(P OR M, R)
= (P OR M). 0
(2.34) Remark. The isomorphism of Corollary 2.33 is given explicitly by
0(f ®9)(P ®m) = f (m)9(p) E R
where f EM',gEP',pEP,andmEM. We will conclude this section by studying the matrix representation of the tensor product of Rmodule homomorphisms. Thus, let R be a commutative ring, let M1, M2, N1, and N2 be finite rank free Rmodules, and let f; : M1 Ni be Rmodule homomorphisms for i = 1, 2. Let m; be the rank of M1 and n; the rank of Ni for i = 1, 2. If M = Ml 0 M2 and N = N10 N2, then it follows from Proposition 2.18 that M and N are free Rmodules of
rank m1n1 and m2n2i respectively. Let f = ft 0 f2 E HomR(M, N). We will compute a matrix representation for f from that for f1 and f2. To do this, suppose that A = {a1 i ... , am, } 5 = {b1, ... , bn, } C = {c1, ... , Cm2 }
D = {d1, ...
, cn2 }
are bases of M1, N1, M2, and N2, respectively. Let E _ {al ®cl, a1 ®c2,... a2 ® C1, a2 ® C2, ...
a1 ®c
,
a2 0 Cm2,
am, ®C1, am, ®C2, ... , amt ® Cm2 } and .2 = {b1 ®dl, b1 ®d2,...
b1 ® dn2,
b2®d1, b2®d2, ... ,b2®42, bn1 ®d1, bn1 ®d2, ... , bn1 ®d2}.
Then E is a basis for M and F is a basis for N. With respect to these bases, there is the following result:
434
Chapter 7. Topics in Module Theory
(2.35) Proposition. With the notation introduced above,
[fl ®f2].y = [fl]B ®[f2]D Proof. Exercise.
l7
Recall that the notion of tensor product of matrices was introduced in Definition 4.1.16 and has been used subsequently in Section 5.5. If [fl]14 = A = [ai3] and (f2]) = B = [,33j], then Proposition 2.35 states that (in block matrix notation)
I (fl ®f2]Y _ an11 B
an12B
There is another possible ordering for the bases C and F. If we set
E'={ai®cj :1 and
i<m1, 1<j<m2}
1"={bi0 dj :1 1 for i > 3 forces t = 4 and d4 = 3.) The three onedimensional representations of G are 7r* (O..) for 0 < i < 2, where Bi are the representations of the cyclic group S constructed in Example 1.4 (6). Now we need to find a threedimensional representation. Let
M=C4={(zl,z2,z3,z4):ziEC, 1 0 EXj(91)Xj(92 1) = EXj(91)Xj(91) =
j=1
since each term is nonnegative and X1(91) = 1 (X1 being the character of the trivial representation r). Then by Corollary 4.16, gl and g2 are conjugate.
0 (4.23) Proposition. Let G be a finite group. Then every complex character of G is real valued if and only if every element in G is conjugate to its oum inverse.
472
Chapter 8. Group Representations
Proof. Suppose that every g is conjugate to g1. Then for every complex character X, X(g) = X(g1) = (g), so X(g) is real. Conversely, suppose that Xi(g) is real for every irreducible complex representation ai and every g E G. Since a1 = r, X1(g) = 1, so e
0
9EG g(W1) is a subrepresentation of V, which must then be V itself). Second, our choice of W1 was arbitrary; we could equally well have chosen some other W. This second observation yields the following useful result:
(5.7) Corollary. Let H be a subgroup of G. Let go E G be fixed and set K = go 1Hgo. Let au : H + Aut(W) be a representation of H, and let aK : Aut(W) be the representation of K defined by aK (9) = aH (90990 1) K for g E K. Then IndH(aH) = IndK(aK). Proof. Let V = IndH (W). Then we may identify W with W, in the statement of Theorem 5.6. Then K = {g E G : g(WW) = Wi} acting on W by the above formula, so V = IndK(W) as well. As a corollary of Theorem 5.6, we may identify two types of representations as induced representations. (5.8) Corollary.
(1) Let V be a transitive permutation representation on the set P = {pi }iEI
and letH={gEG:g(p,)=p,}. Then V=IndH(r). (2) Let V be a transitive monomial representation With respect to the basis
B = {bi}, and let H = {g E G : g9Fbi) = Fb,}. Let a = Fb1 as a representation of H. Then V = Indi(a). Proof.
8.5 Induced Representations
483
Observe from Theorem 5.6 that W is certainly a subrepresentation of ResH IndH(W). We will consider this point in more detail later. However, this observation is enough to enable us to identify some induced representations already. For example, if G = D2m and 0, is one of the 2dimensional representations of G defined in Example 1.4 (7), then each O,, which is a monomial representation, is induced from H = Z,,,, a subgroup of G of index two. If
9, is the representation of that name of H in Example 1.4 (6), i < m/2, This also illusthen IndH(91) _ 0,, and if i > m/2, IndN(91) trates Corollary 5.7, for if g = y (in the notation of Example 1.4 (7)) and (Of all = 8;, then K = H and aK = 9,,,_, and IndX(91) = course, Res$(0,) = 9, 9 9. Let us now concentrate on the case of a normal subgroup H of G and
a representation a : H  Aut(W). If a' is defined by a'(h) = a(ghg1) for some fixed g E G, we call a' a conjugate of a, or more precisely, the conjugate of a by g. Let {aJ } be a complete set of conjugates of a = al . Note that if we let
N(a) _ {g E G : a' : H + Aut(W) defined by a'(h) = a(ghg') is a representation of H isomorphic to a} then N(a) is a subgroup of G containing H and [G: N(a)] is the number of conjugates of a. (Note also that all the subgroups N(a') are mutually conjugate.) The subgroup N(a) is known as the inertia group of a. (5.9) Corollary. Let H be a normal subgroup of G and a : H + Aut(W) a representation of H. Then if a' is any conjugate of a,
Indo(a') = Indy(a). Proof. This is a special case of Corollary 5.7.
0
(5.10) Corollary. Let H be a normal subgroup of G and a : H + Aut(W) a representation of H. Let jail be a complete set of conjugates of a = al. Then
ResH IndH(a) = [N(a) : HI ®,, ai.
Proof. Let {g,} be a set of right coset representatives of H, with gl = 1, and let a, : H  Aut(W) be defined by a,(h) = a(g,'hg,). Then
ResH IndN(a) = ®a, by Theorem 5.6. But in the statement of the corollary we have just grouped
the a, into isomorphism classes, there being [N(a) : H] of these in each class.
0
484
Chapter 8. Group Representations
(5.11) Theorem. (Clifford) Let H be a normal subgroup of G. Let p G  Aut(V) be an irreducible representation of G, and let or : H Aut(W) be any irreducible component of Res°(p). Let Jai) be a complete set of conjugates of or = al, and set Vj = E9ECg(W1), where the sum is taken over the left coset representatives of H such that the conjugate of or
by g is isomorphic to of. Set K = {g E G : g(V,) = Vi}. Then V, is an irreducible representation of K and V = IndK(Vi). Furthermore, K = N(a).
Proof. First consider V' = E9EG g(Wi ). This is an F(G)submodule of V, but V was assumed irreducible, so V' = V. Next observe that instead of summing over g E G, we may instead sum over left coset representatives of H. Further note that the representation of H on g(Wj) is the conjugate of or by g, so we may certainly group the terms together to get V = F, Vi. Now each Vj is a sum of subspaces W;, which are isomorphic to o& as an F(H)module, so by Lemma 7.1.20, each Vj is in fact isomorphic to a direct sum of these. We claim that V = ®j V? . Consider U = V, fl F , V.. Then U is an F(H)submodule of V,, and so is isomorphic to for some ml. Also, U is an F(H)submodule of E»i Vj, and so is isomorphic to e3>, m3W, for some {mj I. By Corollary 7.1.19, ml = m2 = = 0, so U = (0), as required. If a is the conjugate of a by gj E G, then gj(Vl) = V,, so G permutes the subspaces Vj transitively. Then by Theorem 5.6, we obtain V = IndG (Vi).
Also, V, is irreducible, for if it contained a nonzero proper subrepresentation Vi, then IndG(V,') would be a nonzero proper subrepresentation of V, contradicting the irreducibility of V. Finally, g E K if and only if the conjugate by g of W, is isomorphic to W1. But this is exactly the condition for g E N(a). (5.12) Corollary. Let H be a normal subgroup of G. Let p : G + Aut(V) be
an irreducible representation of G, and let or = a' : H  Aut(W,) be any irreducible component of Re$(p). Then (1) (2)
Res$(p) is a semisimple representation of H. In the decomposition of Rest' (p) into a direct sum of irreducible rep
resentations ®mjUj, all of the U, are conjugate to of (and so are
(3)
mutually conjugate), all conjugates of al appear among the U;, and all the multiplicities m3 are equal. Res4(p) = m ®j of for some m E {1, 2, 3, ...} U {oo}, where {a'} is a complete set of conjugates of a1.
8.5 Induced Representations (4)
485
Let H' = {g E G : g(W,) = W,}. Then H C Hl C N(a), and m < [N(a) : H1].
Proof. (1) In the notation of the proof of Theorem 5.11, we have that, as an F(H)module, V = Eg(W1), so V is a sum of simple F(H)modules and hence is semisimple by Lemma 4.3.20. (2) Clearly, all the Uj are conjugate to a1 and all conjugates appear.
Let V, be as in Theorem 5.11, with V, = ®;_i k, (W1) for some group elements ki E G. Then if gj is as in the proof of Theorem 5.11, m1 Wj
Vj
= gi (Vi ) M1
= ®gi(ki(W,)) i=1
m, Wi
so mi = m1 by Corollary 7.1.18. (3) This is merely a restatement of (2). (4) Since g(W1) = W, for g E H', V1 = >J g(W,) where the summa0 tion is over left coset representatives of H1 in N(a).
Remark. It is not true that m = [N(a)
: H1], or even that m divides [N(a) : H'], in general. Let G = Z3 have the representation p = r 9 01
on V=C2 and let H=(1).LetW={(z,z)EC2}. Then N(a)=G, H1=H,so [N(a):H1]=3,but m=2. As a consequence of Clifford's theorem we have the following result:
(5.13) Corollary. Let H be a normal subgroup of G, and let p : G  Aut(V) be an irreducible representation of C. Then if ResH(V) is not isotypic, there is a proper subgroup K of G containing H and an irreducible representation V, of K such that V = IndK(VI).
Proof. Suppose ResH(V) is not isotypic. Let W1 be an irreducible F(H)submodule of ResH (V), and set V, = >g g(W,) where the sum is over all
g E G with g(W,) isomorphic to W1 as an F(H)module, and let K = {g E G : g(V,) = V,}. Then, as above, V1 is an irreducible F(K)module. Note K C H and K 54 G as ResH (V) is not isotypic. Then IndX (V1) is an F(G)submodule of V, but as V is irreducible, it is equal to V. 0 We may use Corollary 5.13 to sharpen Theorem 4.28. (5.14) Theorem. Let G be a finite group and A an abelian normal subgroup of G. If V is an irreducible complex representation of G, then d = deg(V) divides [G : A].
486
Chapter 8. Group Representations
Proof. We prove this by induction on n = IGI. Let V be defined by a : G ' Aut(VG ). Let W = ResA (V ). By Corollary 5.13, if W is not isotypic, V = IndK (V1) for some proper subgroup K of G containing H and some irreducible representation V1 of K. Then deg(V1) divides 1K : H), so deg(V) = [G : K] deg(V1) divides [G : K] [K : H) = [G : H]. If W is isotypic, then W = dW1 for some onedimensional representation W1 of A, or, in other words, a : A Aut(W) = Aut(V) is given by a(g) = multiplication by some complex number X(g) for each g E A. (Also, for any h E G, \(hgh1) = A(g), g E A.) Now for each m consider aem on V ® .. ® V. By Exercise 20, this is an irreducible representation of G"' = G x x G. Let
H is a subgroup of A' and hence a normal subgroup of Gm, and it acts trivially on V®"', so we obtain an irreducible representation of G'IH, which is irreducible as o®m is. Hence, by Theorem 4.28, the degree dm of this representation divides the order of Gl"IH. Note, however, that H has a subgroup
{(gi, ... ,gmi,g1 1 "'9'nl l) : gi E A}
isomorphic to A'"1, so if a = JAI, then d'^ divides nm/a'1 = a(n/a)"' for every m, which implies that d divides n/a = [G: A], as claimed. We now return to the general study of induction. (5.15) Lemma. Let K C H C G be subgroups. (1) (Transitivity of restriction) For any representation V of G,
ResK ResN(V) = ResK(V). (2) (Transitivity of induction) For any representation W of K,
IndN IndK (W) = IndK (W).
Proof. (1) is trivial. As for (2), IndN IndK(W) = F(G) ®F(H) (F(H) ®F(K) W) = (F(G) ®F(H) F(H)) ®F(K) W = F(G) ®F(K) W
= IndK(W)
where the second equality is just the associativity of the tensor product (Theorem 7.2.17).
8.5 Induced Representations
487
The next formula turns out to be tremendously useful, and we will see many examples of its use. Recall that we defined the intertwining number
i(V,W)=(V,W) of two representations in Definition 4.17.
(5.16) Theorem. (Flobenius reciprocity) Let F be an arbitrary field, G an arbitrary group, and H a subgroup of G. Let W be an Frepresentation of H and V an Frepresentation of G. Then (IndH(W), V) = (W, ResH(V)). Proof. By definition,
(IndH(W), V) = dimF Homp(G)(IndH(W), V).
But HomF(G)(IndH(W), V) = HomF(G)(F(G) ®F(H) W, V) = HomF(H)(W,HomF(G)(F(G), V)) = HomF(H)(W, V)
where the second equality is the adjoint associativity of Hom and tensor product (Theorem 7.2.20). Again, by definition, dimFHomF(H)(W, V) = (W, ResH(V)) 0
One direct consequence of Fobenius reciprocity occurs so often that it is worth stating explicitly. (5.17) Corollary. Let F be an excellent field for G. Let W be an irreducible Frepresentation of H and V an irreducible Frepresentation of G. Then the multiplicity of V in IndH (W) is equal to the multiplicity of W in ResH (V). Proof. Immediate from Theorem 5.16 and Lemma 4.18 (3).
0
As an example of the use of Frobenius reciprocity let us use it to provide an alternate proof of part (2) of the fundamental Theorem 3.4. Let F be an excellent field for G and M an irreducible Frepresentation of G of degree d. Let m be the multiplicity of M in F(G). We wish to show m = d. Let H = (1) and recall from Example 5.5 that F(G) = IndH(r). Then
488
Chapter 8. Group Representations
m = (M, F(G)) _ (F(G), M) _ (IndH(T), M) _ (T, ROH(M)) _ (T, dr)
=d as claimed.
(5.18) Example. Consider A4 from Example 3.9, and let F = C. We have a split extension
with V Z2 ® Z2. V has 4 irreducible representations, all of degree 1: T, and three others, which we shall simply denote by A1i A2, and A3. A4 has 4 irreducible representations: 7r'(0i) for i = 0, 1, 2 of degree 1 and a of degree 3. Then (IndV`(T), i'(e:)) _ (T, R.esyA '(ir`(9t))) _ (T, T) = 1,
so 2
Indv°(T) = ®Tr'(9,) i=O
(since
deg(Indv' (T)) = [A4 : V] deg(T) = 3. 1 = 3 and the righthand side is a representation of degree 3). Also, for i = 0, 1, 2
andj=1,2,3
0 = (A,, T) = (Ai, Reso(x'(91))) = (Indy(A,),fr'(9 )), so we must have IndX(A,) = a (as both are of degree 3). Continuing with this example, since we have a split extension, we have a subgroup S of A4 isomorphic to Z3, and we identify S with Z3 via this isomorphism. (We have given S and this isomorphism explicitly in Example 3.9). Now S has three irreducible representations 90 = r, 91 i and 92. Because we have a splitting, R.ess' (7r' (6;)) = 9;, or, more generally,
a,; = (ReSA'(ir'(9,)), 0,) = (ir'(ei), Inds'(91)); since IndA'(91) has degree 4, this gives
Inds'(0,)=7r'(0j)®a
for
i=0, 1,2.
8.5 Induced Representations
489
Here is another useful consequence of Frobenius reciprocity:
(5.19) Proposition. Let G be a group and H an abelian subgroup of G. Let F be an excellent field for G. Then every irreducible Frepresentation of G has degree at most [G : H).
Proof. Let V be an irreducible representation of G. Then ResX (V) is a representation of H and so contains an irreducible representation W, which is onedimensional, since H is abelian. Then
1 < (Res$(V), W) = (V, IndH(W))
the equality being Frobenius reciprocity, so V is a subrepresentation of 0 IndX(W). But deg (IndH(W)) = [G: H], so deg(V) < [G: H]. Example. Note that D2m has an abelian subgroup of index 2 and its irreducible complex representations all have dimension at most 2. The same is true for Q8. Also, A4 has an abelian subgroup of index 3 and its irreducible complex representations all have dimension at most 3. Now we determine the character of an induced representation. In order to state this most simply, we adopt in this theorem the following nonstan
dard notation: If H is a subgroup of G and W an Frepresentation of H with character Xw, we let X'w be the function on G defined by Xw(g) =
f ow(g) if g V H.
(5.20) Theorem. Let H be a subgroup of G of finite index k, and let W be an Frepresentation of H of finite degree. Set V = Indy(W). If {gi}k 1 is a complete set of left coset representatives of H, then for any g E G, k
XV(g)
i.l
Xw(gi 'ggi)
Proof. We know that we may write k
k
V =®gi(W)=®Wi i=1
icl
and that every element of G acts by permuting { Wi }. To be precise, if we let Hi = g1Hgi 1, then
Hi= {gEG:g(W1)=Wi}, Hi, then g(W1) = Wj for some j 0 i. On the other hand, if the representation of Hi on Wi is given by ai : H; ' Aut(Wi), with a, = a, so if g
Chapter 8. Group Representations
490
then for g E H,, a,(9) = a(9; 199i) = a(g'). Now fi : W1  W, by fi(w) = gi(w) is an isomorphism with
(fia(9')f, 1)(wi) = ai(9)(wi)
for allg H, and wiE W,, so ` r(ai(9)) = Tr(a(9 )) Now Xv(g) is the trace of a matrix representing the operation of g. Choose a basis for V that is a union of bases for the Wi. Then if g E G with g 0 Hi, the action of g on W; contributes nothing to Xv(g); while if g E Hi, it contributes Tr(a(9')) = Xw(9') to Xv(g), yielding the theorem. 0
(5.21) Example. In Example 4.29, we found the character table of A5. Let us here adopt an alternate approach, finding the irreducible characters via induced representations. We still know, of course, that they must have degrees 1, 3, 3, 4, 5 and we still denote them as in Example 4.29. Of course, al = r. The construction of a4 was so straightforward that an alternative is hardly necessary, but we shall give one anyway (as we shall need most of the work in any case). Let G = As and let H = A4 included in the obvious way (as permutations of {1, 2, 3, 4) C (1, 2, 3, 4, 5). Then a system of left coset representatives for H is
{1,(12)(45), (12)(35), (13)(25), (23)(15)} = {gi, ... ,g5}. We choose as representatives for the conjugacy classes of C
{1, (14)(23), (123), (12345), (13524)} = {cl, ... ,c5}.
Of course, g' (c1)gi = g, for every i. Otherwise, one can check that 9i 1(cj)gi 0 H except in the following cases: 91 1(c2)91 = c2,
91 1(c3)91 = c3,
and
91 1((123))91 = (132).
(Note that (12 3) = C3 and that while (13 2) is conjugate to it in As, it is not conjugate to it in A4, so we have written the permutation explicitly.) Let W = r and consider IndN (W) = V. Then by Theorem 5.20, it is easy to compute Xv: Xv(c1) = 5,
Xv(c2) = 1,
Xv(C3) = 2,
XV(C4) = XV(C5) = 0
Then 1
(XV,XV)=10(52+15.1+20.22)=2 and r appears in V with multiplicity 1 by Frobenius reciprocity (or by calculating (Xv, X1) = 1), so V contains one other irreducible representation, V = r ® a4, and X4 = Xv  X1. giving the character of a4.
8.5 Induced Representations
491
Now, following Example 3.9, let W = ir'(01) (or 7r'(02)) and consider Indo (W) = V. Again, it is easy to compute Xv: Xv(c1) = 5,
Xv(c2) = 1,
Xv(c3) = exp(21ri/3) +exp(4iri/3) _ 1, XV (C4) = XV (C5) = 0
Now r does not appear in W by Frobenius reciprocity, so that implies here (by considering degrees) that V is irreducible (or alternatively one may calculate that (Xv, XV) = 1) so V = a5 and its character is given above. Now we are left with determining the characters of the two irreducible
representations of degree 3. To find these, let H = Z5 be the subgroup generated by the 5cycle (1 2 3 4 5). Then H has a system of left coset representatives
{1, (14)(23), (243), (142), (234), (143), (12)(34), (13)(24), (123), (134), (124), (132)}
= (91,...,912) Again, g, 1(c1)gj = c1 for every i. Otherwise, one can check that gi 1(cj)gi
H except in the following cases: g11(c4 )91 = c4, 91 1(c5)91 = cs, and
g21((12345))g2 = (15432) _ (12345)1 and gz1((13524))g2 = (14253) = (13524)1. Now let W = 9j and let V = IndH (W). Again by Theorem 5.20 we compute Xv(c1) = 12, Xv(c2) = Xv(c3) = 0, and Xv(c4) = exp(2iri/5) + exp(8ai/5) Xv(c5) = exp(47ri/5) +exp(6ai/5)
if
Xv(c4) = exp(47ri/5) + exp(6ai/5) Xv(c5) = exp(2iri/5) + exp(8iri/5)
if j  2 or 3.
1 or 4,
In any case, one has that r does not appear in V (by either Frobenius reciprocity or calculating (Xv,X1) = 0) and a4 and as each appear in V with multiplicity 1 (by calculating (Xv, X4) = (Xv, Xs) = 1), so their complement is an irreducible representation of degree 3 (which checks with (Xv, Xv) = 3), whose character is Xv  X4  X5Choose j = 1 (or 4) and denote this representation by a3, and choose
j = 2 (or 3) and denote this representation by a3 (and note that they are distinct as their characters are unequal). Then we may calculate that X3(c1) = X3(c,) = 3
X3(2) = X3(2) = 1 X3(C3) = X3(03) = 0
X3(c4) = 1 + exp(2iri/5) + exp(87ri/5) = (1 + f)/2 %3(cs) = 1 + exp(4iri/5) + exp(6ai/5) = (1  f)12 and viceversa for Xs, agreeing with Example 4.29.
Chapter 8. Group Representations
492
Our last main result in this section is Mackey's theorem, which will generalize Corollary 5.10, but, more importantly, give a criterion for an induced representation to be irreducible. We begin with a pair of subgroups K, H of G. A KH double coset is
KgH={kgh:kEK, hEH}. It is easy to check that the KH double cosets partition G (though, unlike for ordinary cosets, they need not have the same cardinality). We shall also refine our previous notation slightly. Let a : H + Aut(W) be a representation of H. For g E C, we shall set H9 = g'Hg and we will let a9 be the representation a9 : H9 . Aut(W) by a9(h) = a(ghg1), for h E H. Finally, let us set H. = H9 fl K. We regard any representation of H, given by a : H  Aut(W), as a representation of H9 by o9 and, hence, as a representation of H. by the restriction of a9 to H9. (In particular, this applies to the regular representation F(H) of H.) (5.22) Theorem. (Mackey) As (F(K), F(H))bimodules,
F(G) ® F(K) ®F(H,) F(H) 9
where the sum is taken over a complete set of KH double coset representatives.
Proof. For simplicity, let us write the righthand side as ®9(F(K)(&F(H))9. Define maps a and f3 as follows:
For g' E G, write g' as g' = kgh for k E K, h E H, and g one of the given double coset representatives, and let a(g') _ (k ® h),. We must
check that a is well defined. Suppose that g' = kgh with k E K and h E H. We need to show (k ®h)9 = (k (& h)9. Now kgh = g' = kgh gives
9_1k'!g = hh 1, and then (k ®h)g = (k(k11) ®(hh1)h)9 ®g_1(klk)9(hh1)h)9 = (k _ (k ®h)9, as required. Then a extends to a map on F(G) by linearity. Conversely, define 3, on K x H by 3.9 (k, h) = kgh E G and extend Q9 to a map
fag : F(K) x F(H) . F(G) by linearity. Then for any x E H9, we have 39(k,9lxgh),
Q9(kx,h) = kx9h = k9(91x9)h =
8.5 Induced Representations
493
so i3g is F(H9)middle linear and so defines
fag : (F(K) (9 F(H))9 ' F(G).
Set Q = rjQg. Then it is easy to check that a and p are inverses of each 0 other, yielding the theorem. Note that the subgroup H. depends not only on the double coset KgH, but on the choice of representative g. However, the modules involved in the statement of Mackey's theorem are independent of this choice. We continue to use the notation of the preceding proof.
(5.23) Proposition. Let g and g be in the same KH double coset. Then
(F(K) ® F(H))g is isomorphic to (F(K) ® F(H))y as (F(K), F(H))bimodules.
Proof. Let g= kgh with k E K, h E H, and define a: K x H K x H by a(k, h) = (kk, hh). Extend to
a : F(K) x F(H) ' F(K) x F(H) by linearity, thus giving
a : F(K) x F(H) + (F(K) 0 F(H))g. We show that a is middle linear: Let x E Hg be arbitrary. Then
a(kx, h) = kxk ®hh = k(xk) ®hh = k ®g'(xk)ghh and
a(k, g'xgh) = kk ®hg'xgh = k ® (9lkg)hg'xgh. But g = kgh, so glkgh(h1g_lk1)x(kgh)h
(9 'kg)hg'xgh = = glxkghh,
and these are equal. Hence, we obtain a map
&: (F(K) (9 F(H))g  (F(K) (& F(H))y. Its inverse is constructed similarly, so i7 is an isomorphism.
0
Remark. Note that a depends on the choice of k and h, which may not be unique, but we have not claimed that there is a unique isomorphism, only that the two bimodules are isomorphic.
494
Chapter 8. Group Representations
(5.24) Corollary. For any Frepresentation W of H, ItesKlndN(W) = ®(F(K) (&F(11,) W) _ ®IndHB(W9). 9
9
Proof. By Theorem 5.22 and associativity of the tensor product (Proposition 7.2.17), we have the following equalities among F(K)modules. IndH(W) = F(G) ®F(H) W
= (®F(K) ®F(H9) F(H)) ®F(H) W 9
_ ® F(K) ®F(H9) (F(H) ®F(H) W) 9
F(K) ®F(H9) W. (Note that although H. is a subgroup of H, its action on F(H) is not the usual action, so one needs to check that F(H) is indeed an (F(H9), F(H))bimodule. but this is immediate.) 0 In the final term in the statement of the corollary, we have denoted the representation by W. to remind the reader of the action of H9. (5.25) Lemma. Let G be a finite group, H a subgroup of G, and F a field of characteristic zero or relatively prime to the order of H. Let W be an Frepresentation of H defined by a : H . Aut(W) and set V = Indy(W). Then
EndG(V) =
HomH9(W9, Re4Hi(W)) 9
where the direct sum is over a complete set of HH double cosets, H9 = H9 n H, and W9 is the representation of H9 on W defined by a9. Proof. The required isomorphism is a consequence of the following chain of equalities and isomorphisms.
Endc(V) = HomG(V, V) = HomH(W1 V)
as in the proof of Frobenius reciprocity L' HomG(V, W) as by our assumption on F, F(H) is semisimple
8.5 Induced Representations
495
= HomF(H) (®F(H) (&F(H9) W, W) 9
= ®HomF(H) (F(H) ®F(H9) W, W) 9
Now note that in the first W above, H9 is operating by o'9; while in the second, it is operating by a _ ® HomF(H9) (W9, ResH9 (W)) 9
by adjoint associativity of Hom and tensor product (Theorem 7.2.20).
11
(5.26) Corollary. (1) Let G be a finite group, H a subgroup of G, and F a field of character
istic zero or relatively prime to the order of H. Let W be a represen
tation of H and set V = IndH(W). Then EndG(V) = F if and only if EndH (W) = F and HomH9 (W9, ResH9 (W)) = 0 for every g E G,
gVH. (2) Let F be an algebraically closed field of characteristic zero or relatively
prime to the order of G. Then V is irreducible if and only if W is irreducible and, for each g E G, g V H, the H9representations W9 and ResH9 (W) are disjoint (i.e., have no mutually isomorphic irreducible components). In particular, if H is a normal subgroup of G, V is irreducible if and only if W is irreducible and distinct from all its conjugates.
Proof. Note Endc(V) contains a subspace isomorphic to EndH(W) (given by the double coset representative g = 1). Also, under our assumptions V 11 is irreducible if and only if EndG(V) = F, and similarly for W.
(5.27) Example. Let H = Z,,, and G = D2,,,,. Then the representations O and 0,,,i of H are conjugate, so for i # m/2, IndH(0i) is irreducible. Of course, this representation is just Oi. (5.28) Example. Let H = A5 and G = S5. As a system of coset representatives, we choose {1, (12 5 4)} = {gl, g2}. Note that
92(12345)g21 = (13524) = (12345)2.
Thus, if we let a1 be the representation a3 of A5 (in the notation of Example 4.29) we see that its conjugate a2 = a3, a distinct irreducible representation. Hence by Corollary 5.26, IndH(a3) = IndH(a') is an irreducible representation of degree 6, and by Theorem 5.20 we may compute its character, giving an alternative to the method of Example 4.30.
496
Chapter 8. Group Representations
(5.29) Example. Let H = V and G = A4 in the notation of Example 5.18. Then the representations A1i A2, and A3 of H are mutually conjugate, so Indy(Ai) is irreducible and is equal to a for i = 1, 2, 3, verifying the result of Example 5.18. (5.30) Example. Let m be such that p = 2'  1 is prime. Then S = Zy acts on V = (Z/2)' by cyclically permuting the elements of V other than the identity. (This may most easily be seen by observing that GL(m, F2) has an element of order p.) Thus, we may form the semidirect product
1iViG*+S+1 with G a group of order n = 2"' (27` 1). G has the onedimensional complex
representations n' (01) for i = 0, ... , p  1. Also, if a is any nontrivial complex representation of V, G has the representation Indy(a) of degree [G : V) = 211. Now a is disjoint from all its conjugates (as Ker(a) may be considered to be an F2vector space of dimension m1, and GL(m1, F2) does not have an element of order p), so by Lemma 5.25, a is irreducible. As (2"  1)2 + 2m1(1)2 = n, these 2' complex representations are all of the irreducible complex representations of G. (Note that if m = 2, then G = A4, so this is a generalization of Example 5.29.)
8.6 Permutation Representations We have already encountered permutation representations, but because of their particular importance, we wish to discuss them here further. We shall restrict our attention to complex representations, so that we may use the full power of character theory. We begin with a bit of recapitulation.
(6.1) Definition. Let P = {pifiEl be a set and a : G
Aut(P) a homo
morphism. Then a defines a representation on CP (=complex vector space with basis P) by
F, a99 9EG
(aiPi) 1: sEl
_
agaia(9)(Pi) g, i
We will often call this representation or as well.
For simplicity, we shall assume throughout that G and P are both finite, though some of our results hold more generally.
8.6 Permutation Representations
497
(6.2) Definition. Let p E P. Then Op = Orbit(p) = {p' E P : v(g)(p) = p' for some g E G}
Gp = Stab(p) = {h E G : a(h)(p) = p}.
Note that Orbit(p) is a subset of P and Stab(p) is a subgroup of G. (Orbit(p) and Stab(p) have been previously defined in Definition 1.4.9.)
(6.3) Definition. A nonempty subset Q of P is a domain of transitivity for
o, if a(g)(q) E Q for every g E G and q E Q. If P is the only domain of transitivity for a (which holds if and only if Op = P for every p E P) then or is called transitive, otherwise intransitive. The following is obvious:
(6.4) Lemma. Let Ql,
... , Qk partition P into domains of transitivity. Then
CP=CQi®...®CQk, Proof.
Recall the following basic result of Corollary 5.8:
(6.5) Theorem. Let a be a transitive permutation representation of G on P. Let p E P and set H = Gp. Then or = IndH(r).
0
Proof.
(Recall that in this situation all of the subgroups Gp, for P E P, are conjugate and we may choose H to be any one of them.) Because of Lemma 6.4, we shall almost always restrict our attention to transitive representations, though we state the next two results more generally.
(6.6) Proposition. Let P be partitioned into k domains of transitivity under the representation a of G. Then the multiplicity of r in CP is equal to k.
Proof. By Lemma 6.4, we may assume k = 1. But then, by Theorem 6.5 and Frobenius recipocity,
(r, v) = (r, Indo(r)) = (Resy(7), r) = 1. 0
Chapter 8. Group Representations
498
We record the following simple but useful result:
(6.7) Lemma. Let ai be a permutation representation of o, on a set Pi for i = 1, 2. Then al and a2 are equivalent if and only if for each g E G, I {pi E Pi : 0`1 (9)(pi)}I = I{p2 E P2 :0.2(9)(P2) =p2}I
Proof. We know that al and 0`2 are equivalent if and only if their characters Xi and X2 are equal. But if X is the character of a permutation representation or of G on a set P, then its character is given by X(9) = I{p E P : a(9)(p) = p}I 0
(6.8) Definition. A partition of P into subsets {Qi}iEr is called a partition into domains of imprimitivity for a if for every g E G and every i E I there
exists a j E I with a(g)(Qi) = Qj. If the only partitions into domains of imprimitivity are either the partition consisting of the single set Q = P, or the partition into subsets of P consisting of single elements, then a is called primitive, otherwise imprimitive. (Note that an intransitive representation of G is certainly imprimitive.) Let G be transitive, Q a subset of P with the property that for every g E G, either a(g)(Q) = Q or a(g)(Q) fl Q = 0. Set
H = {gEG:a(g)(Q)=Q}. Then H is a subgroup of G, and if {gi} are a set of left coset representatives of H, then Qi = gi(Q) partitions P into domains of imprimitivity. Furthermore, all partitions into domains of imprimitivity arise in this way. Note that H acts as a group of permutations on Q. We have the following result, generalizing Theorem 6.5, which also comes from Corollary 5.8.
(6.9) Theorem. Let a be a transitive permutation representation of G on P, Q a domain of imprimitivity for a, and H = {g E G : a(g)(Q) = Q}. If p denotes the permutation representation of H on Q given by p(h)(q) = a(h)(q) for h E H, q E Q, then p is a transitive permutation representation
ofH onQ and a = IndH(p). Proof.
We have the following useful proposition:
(6.10) Proposition. Let a be a transitive permutation representation of G on a set P, and let p E P.
8.6 Permutation Representations
499
(1) Let P be partitioned into a set of domains of imprimitivity {Qi} with
p E Qi, and let H = {g E G : a(g)(Qi) = Qi}. Then there are [G: HI sets in the partition, each of which has [H : Gp) elements. (2) Let H be a subgroup of G containing Gp, and let
Q = {a(g)(p) : g E H}.
Then Q is one element in a partition of G into [G : H) domains of imprimitivity, each of which contains (H : G,J elements.
Proof. Clear from the remark preceding Theorem 6.9, identifying domains 0 of imprimitivity with left cosets.
(6.11) Corollary. A transitive permutation representation a of G on a set P is primitive if and only if some (and hence every) Gp is a maximal subgroup of G. Proof.
0
(6.12) Corollary. Let a be a transitive permutation representation on a set P with IPI prime. Then or is primitive. Proof. By Proposition 6.10, the cardinality of a domain of imprimitivity for G must divide [G : Gp] = JP1, and so consists of either a single element or 0 all of P.
We now introduce another sort of property of a representation.
(6.13) Definition. A permutation representation a of a group C on a set P is kfold transitive if P has at least k elements and for any pair (pl, ... , pk) and (ql, ... , qk) of ktuples of distinct elements of P, there is a g E G with a(g)(pi) = qi for i = 1, ... , k. 2fold transitive is called doubly transitive. (6.14) Examples.
(1) Note that 1fold transitive is just transitive. (2) The permutation representation of D2n on the vertices of an ngon is doubly transitive if n = 3, but only singly (= 1fold) transitive if n > 3. (3) The natural permutation representation of S on {1, ... ,n} is nfold transitive, and of An on 11, ... , n} is (n  2)fold (but not (n I)fold) transitive.
(4) The natural permutation representation of Sn on (ordered or unordered) pairs of elements of {1, ... ,n} is transitive but not doubly
Chapter 8. Group Representations
500
transitive for n > 3, for there is no g E S,, taking {(1, 2), (2,3)1 to {(1,2), (3,4)}. Doubly transitive permutation representations have two useful properties.
(6.15) Proposition. Let a be a doubly transitive permutation representation of G on a set P. Then a is primitive.
Proof. Suppose a is not primitive, and let Q be a domain of imprimitivity, with pi, p2 E Q and p3 0 Q. Then there is no g E G with a(g)(pi) = Pl, and a(g)(p2) = p3, so a is not doubly transitive. The proposition follows by contraposition.
(6.16) Theorem. Let a be a transitive permutation representation of G on
a set P. Then a is doubly transitive if and only if a = r ® a' for some irreducible representation a' of G. Proof. Since a is a permutation representation, its character X is real valued. Note that X2 is the character of the permutation representation a ® a
on P x P. Let k be the number of orbits of a ® a on P x P. Note that k = 2 if a is doubly transitive (the orbits being {(p,p) : p E P} and {(p, q) : p, q E P, p # q}) and k > 2 otherwise. Then, by Lemma 6.4,
k=(a®a,r) 1
= n F, X 2 (9) 9EG
=1
X(g)X(g) gEG
=1
X(g)X(g) gEG
= (X, X).
Note that r is a subrepresentation of a by Lemma 6.4; also note that (X, X) = 2 if and only if in the decomposition of a into irreducibles there are exactly two distinct summands, yielding the theorem. (6.17) Example. The representation 0 of Example 4.29 was doubly transitive and decomposed as r® a4, a4 irreducible. The representation y of Example
4.29 was not doubly transitive; y (D y had three orbits on {1, ... ,5} x 11, ... , 5}, and y decomposed into a sum r ®a4 ED a5 of three distinct irreducibles.
(6.18) Remark. The converse of Proposition 6.15 is false. For example, let p be a prime and consider the permutation representation of D2p on the
8.6 Permutation Representations
501
vertices of a pgon. By Corollary 6.12, this representation is primitive, but for p > 3 it is not doubly transitive.
(6.19) Example. The action of a group on itself by left multiplication is a permutation representation. Of course, this is nothing other than the regular representation, which we have already extensively studied. Instead, we consider the action of a group on itself by conjugation, i.e., y(g)(h) = ghg1 for all g, h E G. Let us determine some examples of this representation. Of course, if G is abelian, this representation is just nr. In any case, the orbits of this representation are just the conjugacy classes. In general, if C1, ... , Ct are the conjugacy classes of G (in some order), we will let ti be
'yonC1,so
7=11®...®7t (1) Consider D2m for m odd. Then (cf. Example 3.7) we have C1 = {1},
C2 = {x, xm1},
... , C(m+l)/2 = {x(m1)/2,x(m+l)/2},
C(m+3)/2 = {y) xy, ... ,
xm1y}.
Then 'y1 = r, and for i = 2, ... , (m + 1)/2, ti is a nontrivial representation containing r as a subrepresentation. Since D2,,, only has one nontrivial onedimensional representation, namely, 0, we see that
y1=rD
...
for
2