A First Course in Module Theory

A First Course in Module Theory

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A First Course in Module Theory

M E Keating

Imperial College, London

ICP

Imperial College Press

Published by Imperial College Press 203 Electrical Engineering Building Imperial College London SW7 2BT Distributed by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Keating, M. E., 1941A first course in module theory / M. E. Keating. p. cm. Includes bibliographical references and index. ISBN 186094096X(alk. paper) 1. Modules (Algebra) QA247.K43 1998 512'.4--dc21

98-9963 CIP

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 1998 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Printed in Singapore by UtoCPdjOyrig/lfec/

Material

To Valerie and Christopher

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Introduction The purpose of this book is to provide an introduction to module theory for a reader who knows something of linear algebra and elementary ring theory. There is a very natural theme for a first course in module theory, namely the structure theory of modules over Euclidean domains. This theory is very explicit, and it has interesting and surprisingly disparate interpretations. An abelian group can be regarded as a module over the ring of integers Z, while a matrix with entries in a field F defines a module over the polyno­ mial ring F[X]. As both Z and F[X] are examples of Euclidean domains, the general theory of modules over Euclidean domains leads to specific re­ sults about abelian groups and about matrices. In the former, we obtain a classification of finitely generated abelian groups, and, in the latter, a description of the rational canonical form and the Jordan normal form of a matrix. Although the structure theory for modules over Euclidean domains is the core of this text, we also consider modules over more general, even noncommutative, rings of coefficients. This extra generality allows us to discuss the limitations and some of the extensions of our main results. The contents of this text are based on a final year undergraduate course that I gave a number of times at Imperial College, London, with some additional material. In the lecture course, I assumed that everyone was familiar with the elementary properties of rings, ideals and Euclidean do­ mains. Here I have provided an introduction to ring theory in the first two chapters, so that the text is more self-contained than the lectures, and a greater variety of rings can be used. Chapters 3 to 7 expound the basics of module theory, including meth­ ods of comparing, constructing and decomposing modules. The results in these chapters are rather general and do not depend much on the ring of coefficients. Chapters 8 to 12 are the heart of this text, since it is here that we obtain the strong results that are special to Euclidean domains. Chapter 12 also contains two applications of the theory, to abelian groups vn

Vlll

Introduction

and to lattices. In Chapter 13, we use the module theory to find two standard forms for a square matrix, namely, the rational canonical form and the Jordan normal form of a matrix. In addition to the usual version of the Jordan normal form of a matrix over the complex numbers, we give two further versions that apply to a matrix whose entries are taken from a field other than the complex numbers. The second of these variations is used, without proof, in a fundamental paper on representation theory by J A Green [Green], and a proof has not been published before at an elementary level, as far as I know. I am grateful to my colleague Gordon James for drawing my attention to these versions of the Jordan normal form. In the final chapter, we go beyond the bounds of Euclidean domains to look at some basic results on projective modules over rings in general. Here, we establish some basic facts about group algebras and the relationship between module theory and the representation theory of groups. As befits a book that is intended as a first course for undergraduates, arguments are given in considerable detail, at least, in the earlier part of text. There is an index entry "proofs in full detail" to help the reader to locate these agruments. There are many explicit illustrations and exercises, and some hints and partial solutions to the exercises are provided in an appendix. Material that was not covered in the original lecture course is indicated C by a "supplementary" symbol as in the margin. This material in not essen­ tial for the core results on modules over Euclidean domains, and it can be omitted if the reader wishes. Each chapter has a section headed "Further developments" in which the definitions and results of the chapter are placed in a wider setting. References are provided to enable the interested reader to follow up the topics that are introduced in these sections. I hope that these sections will provide a useful source of projects for students who are in the final year of a four-year undergraduate MMath or MSci course at a UK university, or who are taking the new one-year preliminary research degree, the "MRes". The text is divided into chapters and sections, which are numbered as you would expect: 1.1, 1.2, and so on. For ease of reference, results are numbered consecutively within each section, and so they appear in the form "12.1.1 Theorem", "12.1.2 Lemma", etc. This book was written while I was also working on the more advanced texts [B & K: IRM] and [B & K: CM] with Jon Berrick of the National Uni­ versity of Singapore. My collaboration with Jon has had many influences, both on my teaching of the lecture course and on the composition of this text, which it is impossible to acknowledge individually.

Introduction

IX

Finally, I should thank the Mathematics Department at Imperial for al­ lowing me the time and space in which to write textbooks. A special thanks also to my colleagues at Imperial for their assistance in mastering IMjrX. Phillip Kent of the METRIC Project ran an introductory course which got me started, and Oliver Pretzel provided me with the LXTJTJX software, and kindly and patiently explained why it sometimes did not do what I hoped.

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Contents Introduction

vii

1 Rings and Ideals 1.1 Groups 1.2 Rings 1.3 Commutative domains 1.4 Units 1.5 Fields 1.6 Polynomial rings 1.7 Ideals 1.8 Principal ideals 1.9 Sum and intersection 1.10 Residue rings 1.11 Residues of integers Exercises

1 1 3 4 4 5 5 7 8 9 10 12 14

2

17 17 18 18 19 20 21 22 23 25 26 28 29 31

Euclidean D o m a i n s 2.1 The definition 2.2 The integers 2.3 Polynomial rings 2.4 The Gaussian integers 2.5 Units and ideals 2.6 Greatest common divisors 2.7 Euclid's algorithm 2.8 Factorization 2.9 Standard factorizations 2.10 Irreducible elements 2.11 Residue rings of Euclidean domains 2.12 Residue rings of polynomial rings 2.13 Splitting fields for polynomials xi

rings

xii

Contents 2.14 Further developments Exercises

31 32

3

Modules and Submodules 3.1 The definition 3.2 Additive groups 3.3 Matrix actions 3.4 Actions of scalar matrices 3.5 Submodules 3.6 Sum and intersection 3.7 fc-foldsums fc-fold sums 3.8 Generators 3.9 Matrix actions again 3.10 Eigenspaces 3.11 Example: a triangular matrix action 3.12 Example: a rotation Exercises

35 35 37 37 39 40 41 43 43 45 46 47 48 48

4

Homomorphisms 4.1 The definition 4.2 Sums and products 4.3 Multiplication homomorphisms 4.4 F[X]-modules in general 4.5 F[X]-module homomorphisms 4.6 The matrix interpretation 4.7 Example: p = 1 4.8 Example: a triangular action 4.9 Kernel and image 4.10 Rank & nullity 4.11 Some calculations 4.12 Isomorphisms 4.13 A submodule correspondence Exercises

51 51 53 54 55 56 57 57 58 58 60 61 62 64 65

5

Free 5.1 5.2 5.3 5.4 5.5 5.6 5.7

69 70 71 73 74 76 77 79

Modules The standard free modules Free modules in general A running example Bases and isomorphisms Uniqueness of rank Change of basis Coordinates

Contents

xiii

5.8 Constructing bases 5.9 Matrices and homomorphisms 5.10 Illustration: the standard case 5.11 Matrices and change of basis 5.12 Determinants and invertible matrices Exercises

80 81 83 84 85 88

6

Quotient Modules and Cyclic Modules 6.1 Quotient modules 6.2 The canonical homomorphism 6.3 Induced homomorphisms 6.4 Cyclic modules 6.5 Submodules of cyclic modules 6.6 The companion matrix 6.7 Cyclic modules over polynomial rings 6.8 Further developments Exercises

91 92 92 93 94 96 99 100 102 102

7

Direct Sums of Modules 7.1 Internal direct sums 7.2 A diagrammatic interpretation 7.3 Indecomposable modules 7.4 Many components 7.5 Block diagonal actions 7.6 External direct sums 7.7 Switching between internal Sz external 7.8 The Chinese Remainder Theorem Exercises

107 107 109 Ill 112 113 114 115 116 118

8

Torsion and the Primary Decomposition 8.1 Torsion elements and modules 8.2 Annihilators of modules 8.3 Primary modules 8.4 The p-primary component 8.5 Cyclic modules 8.6 Further developments Exercises

123 124 125 126 128 130 130 131

9

Presentations 9.1 The definition 9.2 Relations 9.3 Defining a module by relations

133 134 135 136

xiv

Contents 9.4 The fundamental problem 9.5 The presentation matrix 9.6 The presentation homomorphism 9.7 F[X]-module presentations 9.8 Further developments Exercises

136 139 140 141 142 143

10 Diagonalizing and Inverting Matrices 10.1 Elementary operations 10.2 The effect on defining relations 10.3 A matrix interpretation 10.4 Row & column operations in general 10.5 The invariant factor form 10.6 Equivalence of matrices 10.7 A computational technique 10.8 Invertible matrices 10.9 Further developments Exercises

145 145 146 149 150 152 155 156 158 160 161

11 Fitting Ideals 11.1 The definition 11.2 Elementary properties 11.3 Uniqueness of invariant factors 11.4 The characteristic polynomial 11.5 Further developments Exercises

163 163 165 166 167 168 169

12 The Decomposition of Modules 12.1 Submodules of free modules 12.2 Invariant factor presentations 12.3 The invariant factor decomposition 12.4 Some illustrations 12.5 The primary decomposition 12.6 The illustrations, again 12.7 Reconstructing the invariant factors 12.8 The uniqueness results 12.9 A summary 12.10 Abelian groups 12.11 Lattices 12.12 Further developments Exercises

171 171 174 176 178 179 181 182 183 185 186 187 190 190

Contents

xv

13 Normal Forms for Matrices 13.1 F[X]-modules and similarity 13.2 The minimum polynomial 13.3 The rational canonical form 13.4 The Jordan normal form: split case 13.5 A comparison of computations 13.6 The Jordan normal form: nonsplit case 13.7 The Jordan normal form: separable case 13.8 Nilpotent matrices 13.9 Roots of unity 13.10 Further developments Exercises

193 194 195 197 200 202 203 206 209 209 211 211

14 Projective Modules 14.1 The definition 14.2 Split homomorphisms 14.3 Semisimple rings 14.4 Representations of groups 14.5 Hereditary rings Exercises

215 215 216 220 222 224 225

Hints and Solutions

229

Bibliography

243

Index

245

Chapter 1

Rings and Ideals Each module has a ring of scalars associated with it. Therefore, we must discuss rings before we can begin to tackle modules. In this chapter, we collect together the basic definitions and properties of rings that we will need in subsequent chapters. We consider ideals, which tell us about the internal structure of a ring, and we look at some special types of ring, particularly fields and polynomial rings. We also give the construction of residue rings, which is an important method for obtaining new rings from old. A reader who has already met rings, ideals and Euclidean domains may prefer to go directly to the start of our discussion of module theory in Chapter 3, using this chapter and the next for reference. We precede the definition of a ring with the definition of a more funda­ mental structure, namely, a group.

1.1

Groups

We encounter two notations for groups, additive and multiplicative, which are used according to the context in which the group arises. First, we introduce the notation that is most often met in ring theory and module theory. An additive group is a (nonempty) set A together with a law of compo­ sition + , called addition, which behaves as you might expect. Thus for any a, b e A, there is an element a + b £ A which is called the sum of a and b, and the following axioms must be satisfied. 1

Chapter 1. Rings and Ideals

2

Al:

Associativity. (a + b) + c = a + (b + c) for all a, b and c e A.

A2:

Commutativity. a + b = b + a for all a, b € A.

A3: Zero. There is a zero element 0 in A with a + 0 = a for all a in A. A4: Negatives. Each element a oi A has a negative —a so that a + (—a) = 0. It is usual to omit the bracket around a negative and write a + (—b) = a — b and (—a) + b = —a + 6. When studying groups in their own right, the law of composition is usually written in multiplicative notation instead of additive notation. In the multiplicative notation, a group is a (nonempty) set G in which any elements g,h in G have a product g ■ h, often written simply as gh, and the axioms are as follows. Gl:

Associativity. (/ • 9) ■ h = f ■ (g ■ h) for all / , g and

heG.

G2: Identity. There is an identity element 1 in G with g ■ 1 = g = 1 • g for all g in G. G3: Inverses. Each element g of G has an inverse g~l so that S - 9 " 1 = 1 = 0 - 1 •£■ A multiplicative group is abelian or commutative if g ■ h = h- g for all g,h e G. Notice that our list of axioms for a multiplicative group is not simply a translation of the list for an additive group. The difference is found in axiom A2, which demands that an additive group must be abelian. As groups are not our main concern in this text, we will not pause here to give examples of them. We will meet additive groups in profusion in our study of rings and modules, while multiplicative groups will be invoked to construct some examples of rings in Chapter 14.

1.2.

Rings

1.2

3

Rings

Informally, a ring is a set R in which arithmetic can be performed, that is, the members of R can be added and multiplied, in much the same way as integers or real numbers. However, there are two common properties of multiplication that do not hold in a general ring. The first of these properties is that a nonzero element of a ring need not have a multiplicative inverse in that ring. For example, the integer 2 has no inverse within the ring of integers Z, although it does have an inverse in the rational numbers Q. The second property is that multiplication need not be commutative, that is, we can have rs ^ sr for two elements r and s of a ring. Now for our formal definition of a ring. A ring is a nonempty set R, on which there are two laws of composition, addition and multiplication. Addition is indicated by the symbol +, so that for each pair r and s of members of R there is a sum r + s in R. Multiplication is usually indicated by simply writing the elements next to each other: for each pair r,s € R, there is a product rs in R. When it is more convenient to have a symbol for multiplication, we use a dot "•", so the product appears as r ■ s. We use the dot while writing out the axioms. Under addition, R must be an additive group as in the preceding section. The properties of multiplication, and the interaction between addition and multiplication, are given by the following axioms. RM 1:

Associativity. (r ■ s) ■ t = r ■ (s ■ t) for all r, s and t € R.

RM 2: Identity. There is an identity element 1 in R with r ■ 1 = r = 1 • r for all r in R. RM 3:

Distributivity.

For all r, s and t in R, (r + s)-t = r-t + s-t

and r ■ (s + t) = r ■ s + r ■ t.

We allow the possibility that 0 = 1 in a ring. In that event, r = rl = rO = 0 for every r 6 R, so R must be the trivial or zero ring 0 that has only one element. Many statements about rings or modules have trivial exceptional cases when they

Chapter 1. Rings and Ideals

4

s

are interpreted for the zero ring; as a rule, we will not state these exceptions separately. Familiar examples of rings are the ring of integers Z, the ring of rational numbers Q, the ring of real numbers R and the ring of complex numbers C. We shall assume the basic properties of these rings when we need to, as any attempt to establish them in full detail would take us too far from the point of this text. The more leisurely introduction to ring theory given by Allenby [Allenby] does cover these topics. Given a ring R, the set Mn(R) of n x n matrices with entries in R is also a ring under the usual addition and multiplication of matrices. The verification of this assertion is a worthy but lengthy exercise.

1.3

Commutative domains

We now introduce two properties which will be satisfied by many of the rings that we consider in this text. C: A ring R is commutative if rs = sr for all r,s G R. D: A ring R is a domain (alternative names are an integral domain or an entire ring) if R is not the zero ring, and whenever rs = 0 for r,s € R, then either r = 0 or s = 0 already. Familiar examples of domains are the rings Z, Q and R. The standard examples of rings which fail to be commutative or to be domains arise as matrix rings. Take R to be any (nontrivial) ring, Z for instance, and let M2{R) be the set of all 2 x 2 matrices over R. Then M2(R) is neither commutative nor a domain, for if we take r — and s = (

1.4

o o

), then rs =£ 0 but sr — 0.

Units

Let R be a ring. An element u of R is said to be a unit (or invertible) in R if the following condition holds. U: There is an element w in R so that uw = 1 and wu = 1. Such an element w is unique, and it is called the inverse of u. It is usually written u _ 1 or 1/u.

1.5. Fields

5

The inverse u _ 1 of a unit is itself a unit, with inverse u, and the product of two units u, v is again a unit, with inverse v~lu~l. This means that the set U(R) of units in R is a multiplicative group as in section 1.1. Thus U(Z) = { + 1 , - 1 } , while U(Q) is the set of all nonzero rational numbers. Notice that an element that is not a unit in one ring may become a unit in a bigger ring. An important type of ring is defined by the requirement that the zero element is the only non-unit.

1.5

Fields

F. A field is a nonzero commutative ring F in which every nonzero element is a unit of F. The rational numbers Q, the real numbers R and the complex numbers C are all examples of fields. It is a fact that any commutative domain R is contained in a field Q whose elements are of the form r ■ s~x = r/s for r , s € i J , s / 0 . The field Q is called the field of fractions or quotient field of R. For instance, Q is the field of fractions of Z. As the existence of the field of fractions is extremely believable in all the concrete examples that we consider, we will take it for granted. The technical details of the construction are given in several texts - for instance, section 3.10 of [Allenby] or [B & K: IRM] (1.1.12). A discussion of the existence and construction of rings of fractions for more general types of ring can be found in [Rowen] and [B & K: CM], among other texts.

1.6

Polynomial rings

We now introduce rings of polynomials, which play an important role in this text. For the moment, let F be any ring - later F will usually be a field. A polynomial with coefficients in F, or "over F", is an expression / = f(X) = f0 + fxX + f2X2

+■■■+

fmXm

with fo,fi,...,fm € F, where X is an "indeterminate" or "variable". The polynomials / and g = go + giX + g2X2 + ■■■ + gnXn in F[X], with, say, m < n, are equal if /* = #, for i = 1 , . . . , m and 9m+i

= ■ ■ ■ = 9n = 0-

Chapter 1. Rings and Ideals

6

We prefer the notation / to f(X) unless there is a special reason to mention the variable X. If ft = 0 for all t, then / is the zero polynomial 0. If ft = 0 for i > 1, then / is a constant polynomial. When / is not the zero polynomial, the degree of / is the largest index m with fm ^ 0, and we write deg(/) = m. Then fm is the leading term of / . For convenience, the zero polynomial is allocated the degree —oo. We write F[X] for the set of polynomials over F. Addition and multi­ plication of polynomials are defined by the standard rules: given f = fo + fiX + f2X2

+ ■■■ +

fmXm

and 9 = go + giX + g2X2 + ■■■ + gnXn in F[X], with m < n, their sum / + g is given by f + 9 = (fo+9o) + (fi+gi)X

+ --- + (fm + gm)Xm + gm+1Xm+1

and their product fg = f(X)g(X)

+ ---+gnXn

is the polynomial

ho + hxX + ■ ■ ■ + hkXk + ■■■ +

hm+nXm+n

where hk = fo9k H

h fi9k-i H

1- fk9o for k = 0,... ,m +n.

In particular, h0 = fogo, hi = f0gi + figo, and hm+n = fmgn. The verification that F[X] is actually a ring is a matter of careful calcu­ lation. The zero element of F[X] is the zero polynomial, while the identity element is the constant polynomial with /o = 1. Note: our definition of a polynomial is rather informal; for instance we have not defined the "variable" X. A more thorough construction of polynomials is given in section 1.6 of [Allenby]. Our first listed result is elementary but crucial. 1.6.1 Lemma (i) If a ring F is commutative, so also is the polynomial ring F[X]. (ii) If a ring F is a domain, so also is the polynomial ring F[X]. Proof Let f(X) = fo + fiX + hX2 + ■■■ +

fmXm

1.7. Ideals

7

and 9(X) =g0 + giX + g2X2 + ■■■+ gnXn be in F[X], so that fo,fi,...,fm product fg is the polynomial

and go,gi,...,gn

h0 + hiX + --- + hkXk + ■■■ +

are in F; then their hm+nXm+n

with hk = fo9k H 1- fi9k-i H 1- AitoWhen F is commutative, the /c-th coefficient of / # is the same as the fc-th coefficient of gf for k = 0,... ,m + n, since both are the sums of all possible terms ftgk~i, i = 0 , . . . , k, but in different orders. Thus fg = gf for all polynomials / , g, which gives the first assertion. To prove the second, suppose that / and g are both nonzero. We may as well assume that the polynomials have been written so that the coefficients fm and gn are both nonzero. Then the product hm+n = fmgn is also nonzero, so fg is nonzero. □

1.7

Ideals

Ideals are crucial to the investigation of modules. At an elementary level, they provide the first examples of modules, and, at a much deeper level, a knowledge of all the ideals of a ring sometimes enables us to describe the modules over the ring. Ideals come in three types. A left ideal of a ring R is a subset I of R with the following properties. Idl: Zero. The zero element 0 of R is in I. Id2: Additive closure. If x, y € / , then x + y e I. IdL3: Multiplicative closure. If r £ R and x € I, then rx € / . A right ideal I satisfies axioms Idl and Id2, but instead of IdL3 we have IdR3: IfreR and x € I, then xr € / . If I is simultaneously both a left and right ideal of R, we say that I is a two-sided ideal. If R is a commutative ring, then rx = xr for any x and r in R, so that conditions IdL3 and IdR3 are the same. Thus every ideal of R is both left and right and therefore two-sided. In this case, we refer simply to an ideal of R. We will soon see an example in which left and right ideals differ from one another. For any ring R, R itself is a two-sided ideal of R. A (left, right or two-sided) ideal / of R is called proper if I ^ R. The subset {0} of R is a two-sided ideal, called, naturally enough, the zero ideal of R. We usually write 0 for the zero ideal. Next, we see how ideals of a ring arise from the elements of the ring.

8

1.8

Chapter 1. Rings and Ideals

Principal ideals

For any fixed element a of a ring R, let Ra = {ra | r 6 R}. Then Ra is a left ideal of R, called the principal left ideal generated by a. The element a is called a generator of Ra. The principal right ideal generated by a is aR = {ar | r S R}. When R is commutative, Ra = aR is the (two-sided) principal ideal generated by a. The concept of a principal ideal is one of the fundamental notions in this text, since we will usually impose conditions on the ring R which guarantee that all its ideals are principal. To make sure that we get off on the right foot, we give the almost trivial verification that Ra is actually a left ideal. Idl: 0 is in Ra since 0 = Oa. Id2: Suppose that ra and sa are in Ra, where r, s are in R. Then ra + sa = (r + s)a € Ra. Id3: Suppose that r £ R and so € Ra, where s £ R. Then r{sa) = (rs)a is also in Ra. Examples. 1. The principal (left or right) ideal generated by the zero element 0 of R is always the zero ideal, since rO = 0 = Or for every r in R. 2. The principal (left or right) ideal generated by the identity element 1 of R is always R itself, since r l = r = l r for every r in R. 3. In the ring of integers Z, there are ideals 2Z, 3Z, 4Z,....

g

By Lemma 1.8.1, these ideals are all distinct, and, in the next chapter, we see that they are the only proper nonzero ideals of Z (Theorem 2.5.3). Note: our usual rule is that we write a two-sided principal ideal as a left ideal Ra. However, it looks unnatural to write Z2 in place of 2Z, etc. 4. For an example in which left and right ideals differ, we take R to be the ring M 2 (R) of 2 x 2 matrices over the field R of real numbers. Let { 1 0 J. Then 611 = { 0 0

Ren

= {{Z

2) | a i i ' a 2 i G R }

and e

"

R

= { { 7

a

o)\an,a12eRy

1.9. Sum and intersection

9

In the case of greatest interest to us in this text, it is straightforward to determine whether or not two elements of a ring generate the same principal ideal. 1.8.1 Lemma Let R be a commutative domain, and let a and b be nonzero elements of R. Then Ra = Rb if and only if a = ub where u is a unit of R. In particular, Ra = R if and only if a is a unit of R. Proof Suppose that Ra = Rb. Then a = 1 • a is in Rb, so a = ub for some u in R. Similarly, b = wa for some u>. Then a = uwa, and so a(l — uw) = 0. As R is a domain and a ^ 0, we have 1 = uw, which shows that u is a unit with inverse w. Conversely, suppose that a = ub with u a unit. Then ra = (ru)b is in Rb for all r in R, and so Ra C Rb. But b = u~la, so the reverse inclusion also holds. The final assertion is obvious. □

1.9

Sum and intersection

Next we introduce some useful operations on ideals. Suppose that / and J are both left ideals, or both right ideals, or both two-sided ideals of R. Their sum I + J is defined as I + J = {x + y | x € I, y e J}, while their intersection I n J is the usual intersection of sets: 7 n J = { x | a ; € / and x 6 J } . 1.9.1 Lemma If I and J are both left ideals, right ideals or two-sided ideals of a ring R, then I + J and I tl J are also correspondingly left, right or two-sided ideals of R. Proof Suppose that I and J are both left ideals. First, we have 0 e I and 0 e J, giving 0 + 0 = 0 6 / + J . Next, let x + y and x' + y' be members of / + J, with x,x' £ / and y,y' e J. Then (x + y) + ix' + y') = (x + x') + (y + y'),

Chapter 1. Rings and Ideals

10

with x + x' e I and y + y' £ J, so I + J is closed under addition. For any r in R, we have rx £ I and ry £ J. Thus r(x + y) = rx + ry also belongs to J + J, which verifies the final condition. The verification for the intersection is even easier and we leave it to the reader, along with the remaining cases. D

1.10

Residue rings

The construction of a residue ring is a very useful method of obtaining a new ring with interesting properties. We will use this technique to construct some finite fields (Lemma 1.11.2), and to extend a field to a larger field which contains the roots of a given polynomial (Proposition 2.13.1). Let R be a ring and let / be a two-sided ideal of R. Informally, the idea behind the construction of the ring R/I of residues of R modulo I is that two elements r,s of R which differ by an element x in I should give the same element r in R/I. Thus, if r = s + x in R, then f = s in R/I. The point of such a construction can be illustrated by the special case where we take the ideal 2Z in the ring of integers Z. Then two integers r, s differ by an element 2a of 2Z precisely when 2 \r — s, that is, either r and s are both odd or r and s are both even. Thus we expect there to be two members of the residue ring Z/2Z, namely 0 and 1, corresponding in turn to the set of even integers and the set of odd integers. This illustration also explains the use of the term 'residue ring': when an integer is divided by 2, there are two possible 'residues' or 'remainders', 0 or 1. (However, some authors prefer the terms factor ring or quotient ring to residue ring.) Now we turn to the formal construction of the residue ring. Given a two-sided ideal / of a ring R, we define a relationship between the elements of Rby r = s mod / r — s £ I. The above expression is read "r is congruent to s modulo J", and the relationship is called congruence modulo I. When I = Ra is a principal ideal, we usually write r = s mod a rather than r = s mod Ra. Thus for integers r, s, r = s mod 2 r - s £ 2Z 2 | r - s. We record the basic property of congruence. 1.10.1 L e m m a Let I be a two-sided ideal of a ring R. Then congruence modulo I is an equivalence relation on R.

1.10. Residue rings

11

Proof We need to check the three properties which define an equivalence relation, namely, that the given relation is reflexive, symmetric and transi­ tive. For the first, we need to verify that r = r always, which is obvious. The requirement for symmetry is that if r = s, then s = r. But r - s £ I implies s-r e I. For transitivity, we assume that r = s and s = t for three elements r,s,t € R, and we have to show that r = t. But r - 1 = (r - s) + (a -1) e I. D Given an element r of R, we define the residue class of r mod / to be r = {s&R\s

= r mod / } .

Since congruence is an equivalence relation on R, the residue class of r is its equivalence class under this equivalence relation. Thus, by the general properties of equivalence relations, the ring R is partitioned into disjoint residue classes, that is, for r,s G R, either r = s or r P i s = 0. The residue ring R/I is defined to be the set of all residue classes r mod / of elements of R. Thus, as promised, Z/2Z = {0,1}. To make R/I into a ring, we must define addition and multiplication of residue classes. For r,s~ 6 R/I, we put r + ~s = r + s and r • s = rs. The hardest point in the verification that these laws of composition make R/I into a ring is to check that they are well-defined. This problem arises since one element of R/I can be expressed as the residue class of many different elements from R, and we need to know that the sum and product in R/I are not affected by such variations. We record the statement and proof as follows. 1.10.2 Proposition Let I be a two-sided ideal of a ring R. Then (i) addition and multiplication in R/I are well-defined; (ii) R/I is a ring with zero element 0 and identity element 1; (Hi) if R is commutative, then R/I is also a commutative ring. Remark: Exercise 1.8 gives a nontrivial example in which R/I is commu­ tative although R is not.

Chapter 1. Rings and Ideals

12

Proof (i) Suppose that r = ?i and s = si, so that r = n + x and s = si + y for elements x, y € /. Then r + s = (ri+si) + (x + y) with x + y € I and hence r -f s = T\ + Si, that is, the two alternative methods for computing the sum F + s give the same result. Also, rs = r i s i + (r x y + s\x + xy) with rxy + six + xy € I, so the two computations of r • s have the same outcome. (ii) This is a matter of checking that the addition and multiplication in R/I satisfy the axioms as given in section 1.2, granted that the axioms hold in R already. We give two sample checks to show how easy it is. For any r e f l , r+ 0

= =

r +0 f

which shows that 0 is indeed the zero element of R/I. For any r,s,t 6 R, (r + s) -t

= =

r + s -t (r + s)t

=

rt + st

= =

ri + li r -t + s -t

which establishes that the distributive law holds. (iii) Suppose R is commutative. Then, for any elements f,s of R/I, have r-~s=f~s = ~sr = ~s-r.

we

D

1.11

Residues of integers

Our first explicit examples of residue rings arise from the various ideals of the ring of integers Z. The calculations depend on some well-known facts about factorization and division of integers which we take as granted in this chapter. In the next chapter, these facts will be established as special cases of results that hold for Euclidean domains in general. Let m be a positive integer and let / = mZ be the principal ideal generated by m. We introduce the special notation Z m for the ring Z/mZ, since these rings appear frequently in this text.

1.11. Residues of integers

13

To describe Z m , we use the fact that, for any integer s, there are integers q and r with s = qm + r and 0 < r < m — 1. Now, for integers r, s, s = f in Z m s = r mod m s — r € mZ m | s — r. Thus the set of residue classes mod m is Z m = { 0 , T , . . . , m - 1}, and these are all distinct since m cannot divide r — s if 0 < r, s < m — 1 unless r = s. Note that the ring Z m need not be a domain even though Z is a domain - for example 2 • 2 = 0 in Z4, although 2 ^ 0 . However, in the important case when p is a prime number, the ring Z p is a field, which we deduce from a more general result that is our next theorem. Before we can state this theorem, we need a preliminary definition. A two-sided ideal I of a ring R is maximal if I is a proper ideal of R, and there is no two-sided ideal J of R with I C J C R. Here, we use the symbol "C" to indicate strict containment, that is, I ^ J and J / R. Maximal left ideals and maximal right ideals are defined in the obvious way. Granted the unique factorization of integers, and that every ideal in Z is principal, the maximal ideals in Z have the form pZ where p is prime. 1.11.1 T h e o r e m Let I be an ideal of a commutative ring R. ments are equivalent.

Then the following state­

(i) I is a maximal ideal of R. (ii) The residue ring R/I is a field. Proof (i) => (ii): By Lemma 1.10.2, R/I is a commutative ring, so we only need to find an inverse for a typical nonzero element f € R/I. By Lemma 1.9.1, Rr+I is an ideal of R, and, as / is maximal, either Rr+I = I or Rr+I = R. But if Rr + I = I, then r = l-r + 0el and so f = 0, contradicting our assumption that r is nonzero. Thus Rr + I = R, and so there is an element s of R and an element x of I with sr + x = 1. This equality gives s • f = 1 in R/I, that is, f has an inverse. (ii) => (i): Suppose that J is an ideal of R with I C J, (so I ^ J). Then there is some element r £ J with r $ I. Since F ^ 0 in R/I, s • f = 1 for some s € R.

Chapter 1. Rings and Ideals

14

But then 1 = sr + x in R for some element x of J, and so 1 £ J, which implies that J = R. Thus / is a maximal ideal of R, as desired. □ 1.11.2 Corollary Let p be a prime number. Then Z p is a

field.

D

Exercises 1.1

1.2 1.3

1.4

1.5

1.6

g

1.7

A nonzero element r of a ring R is called a proper zero divisor if rs = 0 for some nonzero element s £ R. Show that a proper zero divisor cannot be a unit of R. Show that a commutative ring R is a field if and only if 0 is the only proper ideal of R. Let R be a commutative ring. An ideal / of R is said to be prime if the following holds. P: If r, s £ R and rs £ I, then either r £ I or s £ I (possibly both are in / ) . Show that (i) 0 is a prime ideal of R «=>• R is a domain. (ii) / is prime R/I is a domain. (iii) A maximal ideal of R must be a prime ideal. Let m > 2 be an integer. Show that Z m is a domain • m is a prime number. Show by direct calculation that U(ZQ) = {1,5} and that the set of proper zero divisors in I,Q is {2,3,4}. Find the corresponding results for Zg and ZioLet A be a commutative domain and let R = A[X] be the polynomial ring over A. Prove the following assertions. (a) / = f{X) = /o + fiX + ■ ■ • + fnXn is a unit of R n = 0 and /o is a unit of A. (b) X divides a product fg in i? ■ X divides either / or g. (c) X = fg with f,g £ R «=*> * either / or g is a unit of R. Let F be a field and let fl = F[X, Y] (= F[X][Y]) be the polynomial ring in two variables over F. Show the following. (a) R is a domain. (b) RX is a prime ideal of R. (c) i?X + RY is a maximal ideal of R but .RX + RY is not principal. Let F be a field and let D be the set of all diagonal matrices ( ^

]

with r,s € F. Verify that D is a commutative ring under the usual sum and product of matrices.

Exercises

15 Show also that the only proper nonzero ideals of D are De and

Df, where e =(

J

jj J and / = ( J

J J, and that De + D / = D

and DedDf = 0. Show that there is a bijective map 9 : D/De —► F given by r

0

x 0

s

and that 9{d + d1) = 9{d) + 6(d') and 6(d ■ df) = 9(d) ■ 9(d')

1.8

for all d, d' in D. Thus we can identify the ring D/De with F. Remark in other words, 9 is an isomorphism of rings - see Exercise 4.4. Let F be a field and let T be the set of all upper triangular matrices C

/ r

t \

I n . ) with r, s,t in F. Verify that T is a ring under the usual sum and product of matrices. Show that T is not commutative. Let

H

$) | r , t 6 i r }

={{I 0 0

t s

t,seF

'-{(SO1"'} 1.9

Show that i7, J and J are all two-sided ideals in T. Using the methods of the preceding exercise, show that T/H = T/I = F, while T/J = D. (Thus, as promised earlier, a noncommutative ring can have a commutative residue ring.) Let F be a field and let R be the ring of all 2 x 2 matrices over F . Show that R has no two-sided ideals except 0 and R. (See Exercise 7.8.)

Chapter 2

Euclidean Domains We now introduce the type of ring which is of greatest interest to us in this text, namely, a Euclidean domain. Such a ring shares with the integers Z the property that long division is possible, that is, given elements a, b of the ring, then there are elements q, r of the ring with a = qb + r where the remainder r is 'smaller' than b in some sense. Consequently, many results that hold for the ring of integers can be extended to Euclidean domains in general. Apart from the integers themselves, most of Euclidean domains that we encounter in this text are polynomial rings of the form F[X] for a field F. We briefly consider the Gaussian integers Z(»], and some further examples are mentioned in the exercises. This chapter also contains a detailed analysis of the residue rings of polynomial rings. This analysis is used immediately to give an algebraic method for constructing roots of polynomials, and later, in Chapter 13, to find normal forms for matrices.

2.1

The definition

A Euclidean domain is a commutative domain R together with a function ip : R —► Z that has the following properties. ED 1: ip(a) > 0 for all r € R, and a = 0. ED 2: 1 and we may as well renumber the indices so that h = 2. But then (X — Xi){X — A2) has real coefficients and it is a factor of / . Thus / = (X - Xx)(X - A2). Writing b = —(Ai + A2) and c = A1A2 gives the required form for / .

Irreducible rational polynomials There is no general description of the irreducible polynomials over Q[X]. (If there were, algebraic number theory would be an easier subject!) We quote two useful results for handling rational polynomials; the first is proved in several of our references: [Allenby], [Cohn 1] and [Marcus], and the second is a not-too-difficult consequence - see Exercise 2.3. Gauss' Lemma. Suppose that / is a monic polynomial with integer coefficients and that / = gh with / , g £ Q[X] both monic. Then g and h also have integer coefficients. Eisenstein's Criterion. Let / = Xn + / n _ i X n _ 1 + • • • + f\X + f0 be a monic polynomial with integer coefficients, and suppose that there is a prime number p such that p \ fi for i = 0 , . . . , n — 1 but p2 does not divide /o. Then / is irreducible over Q. These results remain true if Q is replaced by the field of fractions Q of a Euclidean domain R and p is taken to be an irreducible element of R.

28

Chapter 2. Euclidean Domains

Irreducible polynomials over finite fields Although there are irreducible polynomials of every degree over a finite field ([Allenby] p. 163), the argument that predicts their existence is indirect and does not tell us what they look like. For small fields, it is possible to list the irreducible polynomials of a given small degree by enumerating those that are reducible. For example, take F = 1>2 = {0,1} - here, it is convenient to omit the "bars" that indicate we are working with residue classes. There are two linear monic polynomials in Z2[X], namely X and X + 1, and four quadratic monic polynomials, X2, X2 + 1, X2 + X and X2 + X + 1. The first two are squares (note that 2 = 0 in Z2) and the third is a product. Thus X2 + X + \ is, by elimination, the only possible irreducible polynomial of degree 2 over Zg. It actually is irreducible, since neither element of Z2 is a root.

2.11

Residue rings of Euclidean domains

The fact that every ideal of a Euclidean domain is principal, together with the Unique Factorization Theorem, leads to a good description of the residue rings of a Euclidean domain. We will postpone the general treatment of this topic until we discuss cyclic modules in Chapter 6, since the discussion can be simplified when we have some machinery from mod­ ule theory at our disposal. In the remainder of this chapter, we will show how fields arise as residue rings of Euclidean domains and we will give an explicit description of the residue rings of polynomial rings. Combining these, we obtain an algebraic construction for roots of polynomials. First, we show how fields arise. 2.11.1 Proposition Let R be a Euclidean domain, I an ideal of R. Then the residue ring R/I is a field