320
;" .
;" .
A First Course in RINGS AND IDEALS
. DAVID M.~BURTON University ofNew Hampshire
.' 'Y 'Y - ADDISON-WESLEY PUBLISmNG COMPANY
Reading, Massachusetts . Menlo Park, California . London . Don MilIs, Ontario
;" .
A First Course in RINGS AND IDEALS
. DAVID M.~BURTON University ofNew Hampshire
.' 'Y 'Y - ADDISON-WESLEY PUBLISmNG COMPANY
Reading, Massachusetts . Menlo Park, California . London . Don MilIs, Ontario
This book .s in the ADDISON-WESLEY SERIES IN MATHEMATICS
Consulting Editor: Lynn H. Loomis
Standard Book Number 201.00731.2 AMS 1968 Subject Classifieations 1610, 1620. Copyright 1970 by Addison-Wesley Publishing Company, lnc. Philippines copyright 1970 by Addison~Wesley PubJishing Company, Ine. AII rights reserved. No part ofthis publieation may be reproduced. stored in a retrievaI system, or transmitted. in any form or by any means. electronie, mechanical, photocopying, recording. or otherwise, without the prior written permission of the publisher. Printed in Ihe United Sta tes , of Ameriea. Published simultaneous1y in Canada. Libraryof Congress CataJog Card No. 73-100855.
.~ '{ I
1 '1 '.1 l I
1 i
1
To my Father. Frank Howard Burton
This book .s in the ADDISON-WESLEY SERIES IN MATHEMATICS
Consulting Editor: Lynn H. Loomis
Standard Book Number 201.00731.2 AMS 1968 Subject Classifieations 1610, 1620. Copyright 1970 by Addison-Wesley Publishing Company, lnc. Philippines copyright 1970 by Addison~Wesley PubJishing Company, Ine. AII rights reserved. No part ofthis publieation may be reproduced. stored in a retrievaI system, or transmitted. in any form or by any means. electronie, mechanical, photocopying, recording. or otherwise, without the prior written permission of the publisher. Printed in Ihe United Sta tes , of Ameriea. Published simultaneous1y in Canada. Libraryof Congress CataJog Card No. 73-100855.
.~ '{ I
1 '1 '.1 l I
1 i
1
To my Father. Frank Howard Burton
~' el .~
,
----------------------~~ ----,------------ .----_.-.. _.- ---- ----:-----.---------- --"----- _._----_.,
:1'
.' ".
PREFACE
As the title suggests, this volume is designed to 'serve as an introduction to the basic ideas and techniques of ring theory; it is intendedto be an expository textbook, rather than a treatise on the subject. Th,mathe-matical background required for a proper understanding of the coi1tents
i~ not extensive. We assume that the average reader 4as had SbITle prior contact with' abstract algebra, but is stll relatively inexperienced";in this respect. In consequence, nearly everything herein can be read by a person familiar with basic group-theoretic concepts and having a nodding acquain-tance with linear algebra; .
The level ofmaterial should p'rove suitable-for advanced undergraduates and beginning graduate students. Indeed, a built-in flexibility perrnits the book to be used, either as the basic text or to be read independently by interested students, in a variety of situations. The reader whose main interest is in ideal theory, for instance, could chart a course through Chapters 2, 3, 5, 8, 11, 12, 13. Taken as a whole, the present work is more nearly a begin-ning than an end. Our hope is that it may serve as a n~tural point of departure for the study of th,e advanced treatises on ring theory and, in sorne aspects of the subject, the periodicalliterature.
As regards treatment, ollr guiding principIe is the'sifong conviction that intelligibility should be given priority over coverage; that a deeper under-standing of a few important topics is preferable to a superficial knowledge of many. This calIs for 'a presentation in which the pace is unhurried and which is complete in the detaiis of proof, particularIy of basic resuIts. By adhering to the "theorem-proof" ,5tyle ofwriting, we hope to achieve greater clarity (perhaps at the sacrifice of elegance). Apart from the general know-ledge presupposed, an attempt has been made to keep the text technicalIy self-contained, even to the extent of including sorne material which is undoubtedly familiar. The mathematialIy sophisticated reader may prefer to skip the earlier chapters and refer to them only if the need arises.
At the end of each chapter, there will be found a colIection of problems of varying degrees of difficulty. These constitute an integral part of the book and reqliire the reader's active participation. They introduce a variety
v
~' el .~
,
----------------------~~ ----,------------ .----_.-.. _.- ---- ----:-----.---------- --"----- _._----_.,
:1'
.' ".
PREFACE
As the title suggests, this volume is designed to 'serve as an introduction to the basic ideas and techniques of ring theory; it is intendedto be an expository textbook, rather than a treatise on the subject. Th,mathe-matical background required for a proper understanding of the coi1tents
i~ not extensive. We assume that the average reader 4as had SbITle prior contact with' abstract algebra, but is stll relatively inexperienced";in this respect. In consequence, nearly everything herein can be read by a person familiar with basic group-theoretic concepts and having a nodding acquain-tance with linear algebra; .
The level ofmaterial should p'rove suitable-for advanced undergraduates and beginning graduate students. Indeed, a built-in flexibility perrnits the book to be used, either as the basic text or to be read independently by interested students, in a variety of situations. The reader whose main interest is in ideal theory, for instance, could chart a course through Chapters 2, 3, 5, 8, 11, 12, 13. Taken as a whole, the present work is more nearly a begin-ning than an end. Our hope is that it may serve as a n~tural point of departure for the study of th,e advanced treatises on ring theory and, in sorne aspects of the subject, the periodicalliterature.
As regards treatment, ollr guiding principIe is the'sifong conviction that intelligibility should be given priority over coverage; that a deeper under-standing of a few important topics is preferable to a superficial knowledge of many. This calIs for 'a presentation in which the pace is unhurried and which is complete in the detaiis of proof, particularIy of basic resuIts. By adhering to the "theorem-proof" ,5tyle ofwriting, we hope to achieve greater clarity (perhaps at the sacrifice of elegance). Apart from the general know-ledge presupposed, an attempt has been made to keep the text technicalIy self-contained, even to the extent of including sorne material which is undoubtedly familiar. The mathematialIy sophisticated reader may prefer to skip the earlier chapters and refer to them only if the need arises.
At the end of each chapter, there will be found a colIection of problems of varying degrees of difficulty. These constitute an integral part of the book and reqliire the reader's active participation. They introduce a variety
v
o'
7 I
vi PREFACE
of topics not treated in the body of the text, as well as impart additional information about material covered earlier; sorne, especially in the later chapters, provide substantial extensions of the theory. We have, on the whole, resisted the temptation to use the exercises to develop results that will subsequently be needed (although this is not hard and fast). Those problems whose solutions do not appear straightforward are often accom-panied by hints.
The text is not intended to be encyc10pedic in nature; many fascinating aspects of this subject vie for inc1usion and sorne choice is imperative. To this end, we merely followed our own tastes, condensing or omitting entirely a number of topics that might have been encompassed by a book of the same tltle. Despite sorne notable omissions, the coverage should provide a firm foundation on which to build.
A great deal of valuable criticism was received in the preparation of this work and ourmoments of complacence have admitted many improvements. Of those students who helped, consciously or otherwise, we should like particularly to mention Francis Chevarley, Delmon Grapes, Cynthia Kennett, Kenneth Lidman, Roy Morell, Brenda Phalen, David Smith, and John Sundstrom; we valued their critical reading of sections of the manu-script and incorporated a number of their suggestions into the texto It is a pleasure, likewise, to record our indebtedness to Professor James Clay of the University of Arizoria, who reviewed the final draft and offered helpful comments leading to its correction and improvement. We also profited from many conversations with our colleagues at the University of New Hampshire, especial1y Professors Edward Batho, Homer Bechtell, Robb Jacoby, and Richard Johnson. In this regard, special thanks are due Pro-fessor William Witthft, who was kind enough to read portions of the galleys; his eagle-eyed attention saved us from embarrassment more than once. We enjoyed the'luxury of unusually good secretarial help and take this occasion to express our appreciation to Nancy Buchanan and Sola'nge Larochelle for their joint labors on the typescript. To my wife must go tbe largest .debt of gratitud e, not only for generous assistance with the text at all stages of development, but for her patience and understanding on those occasions when nOtlling would go as we wished.
Finally, we should like to acknowledge the fine cooperation of the staff of Addison-Wesley and the usual high quality of their work. The author, needless tq say, must accept the full responsibility for any shortcomings and errors which remain.
Durham, New Hampshire J anuary 1970
D.M.B.
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter lQ Chapter 11
Chapter 12
Chapter 13
C;rl: 4 585
CONTENTS
Introductory Concepts .
Ideals and Their Operations 16
The Classical Isomorphism Theorems . 39
Integral Domains and Fields 52
Maximal, Prime, and Primary Ideals 71
Divisibility Theory in Integral Domains 90
PolynomiaI Rings 112 ...
Certain Radicals of a Ring . 157
Two Classic Theorems 180
Direct Sums of Rings 204
Rings with Chain Conditions 217
Further Results on Noetherian Rings . 234
Some Noncommutative Theory 262
AppendixA. Relations. 287
AppendixB. Zorn's Lernma 296
, Bibliography 300
Index oC Special Symbols 303
Index . 305
vii
o'
7 I
vi PREFACE
of topics not treated in the body of the text, as well as impart additional information about material covered earlier; sorne, especially in the later chapters, provide substantial extensions of the theory. We have, on the whole, resisted the temptation to use the exercises to develop results that will subsequently be needed (although this is not hard and fast). Those problems whose solutions do not appear straightforward are often accom-panied by hints.
The text is not intended to be encyc10pedic in nature; many fascinating aspects of this subject vie for inc1usion and sorne choice is imperative. To this end, we merely followed our own tastes, condensing or omitting entirely a number of topics that might have been encompassed by a book of the same tltle. Despite sorne notable omissions, the coverage should provide a firm foundation on which to build.
A great deal of valuable criticism was received in the preparation of this work and ourmoments of complacence have admitted many improvements. Of those students who helped, consciously or otherwise, we should like particularly to mention Francis Chevarley, Delmon Grapes, Cynthia Kennett, Kenneth Lidman, Roy Morell, Brenda Phalen, David Smith, and John Sundstrom; we valued their critical reading of sections of the manu-script and incorporated a number of their suggestions into the texto It is a pleasure, likewise, to record our indebtedness to Professor James Clay of the University of Arizoria, who reviewed the final draft and offered helpful comments leading to its correction and improvement. We also profited from many conversations with our colleagues at the University of New Hampshire, especial1y Professors Edward Batho, Homer Bechtell, Robb Jacoby, and Richard Johnson. In this regard, special thanks are due Pro-fessor William Witthft, who was kind enough to read portions of the galleys; his eagle-eyed attention saved us from embarrassment more than once. We enjoyed the'luxury of unusually good secretarial help and take this occasion to express our appreciation to Nancy Buchanan and Sola'nge Larochelle for their joint labors on the typescript. To my wife must go tbe largest .debt of gratitud e, not only for generous assistance with the text at all stages of development, but for her patience and understanding on those occasions when nOtlling would go as we wished.
Finally, we should like to acknowledge the fine cooperation of the staff of Addison-Wesley and the usual high quality of their work. The author, needless tq say, must accept the full responsibility for any shortcomings and errors which remain.
Durham, New Hampshire J anuary 1970
D.M.B.
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter lQ Chapter 11
Chapter 12
Chapter 13
C;rl: 4 585
CONTENTS
Introductory Concepts .
Ideals and Their Operations 16
The Classical Isomorphism Theorems . 39
Integral Domains and Fields 52
Maximal, Prime, and Primary Ideals 71
Divisibility Theory in Integral Domains 90
PolynomiaI Rings 112 ...
Certain Radicals of a Ring . 157
Two Classic Theorems 180
Direct Sums of Rings 204
Rings with Chain Conditions 217
Further Results on Noetherian Rings . 234
Some Noncommutative Theory 262
AppendixA. Relations. 287
AppendixB. Zorn's Lernma 296
, Bibliography 300
Index oC Special Symbols 303
Index . 305
vii
.,'
CONVENTIONS
Rere we sha11 set forth certain conventions in notatio'i(,~nd terminology used throughout. the text: the standard sym bols of se" theory will be
e~ployed, namely, E, u, n, -, and 0 for the empty set. In particular, A - B = {xix E A and x! B}. As regards inclusion, the symbols ~ ~~d ;;2 mean ordinary inclusion between sets (they do not exclude the posslbIllty of equality), whereas e and ::J indicate proper inclusion. When we ~eal with an indexed co11ection of sets, say {Ali E IJ, the cumbersome notatlOns U {AliEI} and n {AliEI} will genera11y. be ~bbreviated to. u A and n A; it being uIiderstood that the operabons are always over the fu11 domain on which the index is defined. Fo11owing custom, {a} denotes the set whose only member is a. Provided that there is no dsk of confusion, a one-element set will be identified with the element itself. .
A function f (synonymous with mapping) is indicated by a strai?ht arrow going from domain to range, as in the case f: X .-+ Y, and the no.tatl~n always signifies thatfhas domain X. Under these cl~cumstan.ces,fls sald to be a function on X, or from X, into. Y. In representmg functlOnal values, we adopt the convention of writing the .function on the left, so that/~x), or occasiona11y fx, denotes the image of an element x E X. The restnctlOn of f to a subset A of X is the function flA from.1 into Y defined. by (fIA)(x) = f(x) for a11 x in A: F~r the compo.sltlOn of two func~lOns f: X -+ Yand g: Y -+ Z, we wIlI wnte g o f; that lS, g o f: X ~ Z .satlsfies (g o f)(x) = g(J(x)) for each x E X. (It is important to bear m mmd that our policy is to apply the functions from right to left.)
Sorne knowledge of elementary number theory is assumed. We simply remark that the term "prime number" is taken to mean a positive prime; in other words, an integer n > 1 whose only divisors are 1 and n. Fina11y, let us reserve the symbol Zfor th~ set of all integers, Z~ for the set of positive integers, Q fo! the set of ratlOnal numbers, and R for the set of real numbers.
viii
.ONE
INTRODUCTORY CONCEPTS
The present chapter sets the stag'~for much that fo11ows, by reviewing sorne of the basic elements of ring theory: I t al so serves as an appropriate vehicle for codifying certain notation and, technical vocabulary used throughout the text . With an eye to the b~,r~ning student (as well as .to minimize a sense of vagueness), we have also'l!1duded a ~umbel of pertinenrexamples of rings. The mathematica11y matre reader who finds thepace'somewhat tedious may prefer to bypass this section, referring to it for terminology w hen necessary.
As a starting point, it would seem appropriate formally to define the principal object of inten!st in this book, the notion of a ringo
Definition 1-1. A ring is an ordered triple (R, +,.) consisting of a nonempty set R and two binary operations + and . defined on R such that
1) (R, +) is a cornmutative group, 2) (R,') is a semigroup, and 3) the operation . is distributive (on both sides) over the operation +.
The reader should understand clearly that +' and . represent abstract, unspecified, operations and not ordinary addition and multiplication. For convenience, however, one invariably refers to the operation + as addition and to the operation . as multiplication. In the light of this terminology, it is natural then to speak of the commutative group (R, +) as the additive group of the ri.p.g and of (R, .) as the multiplicative semigroup of the. ringo
By analogy with the integers, the unique identity element for addition is caBed the zero element of the ring and is denoted by the usual symbol O. The unique additive inverse of an element a E R will hereafter be written as - a. (See Problem 1 for justification of the adjective "unique".)
In order to minimize the use of parentheses in expressions involving both operations, we shall stipulate that multiplication is to be performed befo re addition. Accordingly, the expression a'b + e stand s for (a'b) + e and not for a'(b + e). Because ofthe general associative law, parentheses
.,'
CONVENTIONS
Rere we sha11 set forth certain conventions in notatio'i(,~nd terminology used throughout. the text: the standard sym bols of se" theory will be
e~ployed, namely, E, u, n, -, and 0 for the empty set. In particular, A - B = {xix E A and x! B}. As regards inclusion, the symbols ~ ~~d ;;2 mean ordinary inclusion between sets (they do not exclude the posslbIllty of equality), whereas e and ::J indicate proper inclusion. When we ~eal with an indexed co11ection of sets, say {Ali E IJ, the cumbersome notatlOns U {AliEI} and n {AliEI} will genera11y. be ~bbreviated to. u A and n A; it being uIiderstood that the operabons are always over the fu11 domain on which the index is defined. Fo11owing custom, {a} denotes the set whose only member is a. Provided that there is no dsk of confusion, a one-element set will be identified with the element itself. .
A function f (synonymous with mapping) is indicated by a strai?ht arrow going from domain to range, as in the case f: X .-+ Y, and the no.tatl~n always signifies thatfhas domain X. Under these cl~cumstan.ces,fls sald to be a function on X, or from X, into. Y. In representmg functlOnal values, we adopt the convention of writing the .function on the left, so that/~x), or occasiona11y fx, denotes the image of an element x E X. The restnctlOn of f to a subset A of X is the function flA from.1 into Y defined. by (fIA)(x) = f(x) for a11 x in A: F~r the compo.sltlOn of two func~lOns f: X -+ Yand g: Y -+ Z, we wIlI wnte g o f; that lS, g o f: X ~ Z .satlsfies (g o f)(x) = g(J(x)) for each x E X. (It is important to bear m mmd that our policy is to apply the functions from right to left.)
Sorne knowledge of elementary number theory is assumed. We simply remark that the term "prime number" is taken to mean a positive prime; in other words, an integer n > 1 whose only divisors are 1 and n. Fina11y, let us reserve the symbol Zfor th~ set of all integers, Z~ for the set of positive integers, Q fo! the set of ratlOnal numbers, and R for the set of real numbers.
viii
.ONE
INTRODUCTORY CONCEPTS
The present chapter sets the stag'~for much that fo11ows, by reviewing sorne of the basic elements of ring theory: I t al so serves as an appropriate vehicle for codifying certain notation and, technical vocabulary used throughout the text . With an eye to the b~,r~ning student (as well as .to minimize a sense of vagueness), we have also'l!1duded a ~umbel of pertinenrexamples of rings. The mathematica11y matre reader who finds thepace'somewhat tedious may prefer to bypass this section, referring to it for terminology w hen necessary.
As a starting point, it would seem appropriate formally to define the principal object of inten!st in this book, the notion of a ringo
Definition 1-1. A ring is an ordered triple (R, +,.) consisting of a nonempty set R and two binary operations + and . defined on R such that
1) (R, +) is a cornmutative group, 2) (R,') is a semigroup, and 3) the operation . is distributive (on both sides) over the operation +.
The reader should understand clearly that +' and . represent abstract, unspecified, operations and not ordinary addition and multiplication. For convenience, however, one invariably refers to the operation + as addition and to the operation . as multiplication. In the light of this terminology, it is natural then to speak of the commutative group (R, +) as the additive group of the ri.p.g and of (R, .) as the multiplicative semigroup of the. ringo
By analogy with the integers, the unique identity element for addition is caBed the zero element of the ring and is denoted by the usual symbol O. The unique additive inverse of an element a E R will hereafter be written as - a. (See Problem 1 for justification of the adjective "unique".)
In order to minimize the use of parentheses in expressions involving both operations, we shall stipulate that multiplication is to be performed befo re addition. Accordingly, the expression a'b + e stand s for (a'b) + e and not for a'(b + e). Because ofthe general associative law, parentheses
2 FIRST COURSE IN RINGS AND IDEALS
can also be otntted when writing out sums and products of more than two elements.
With these remarks in mind, we can now give a more elaborate definition of a ringo A ring (R, +, . ) consists of a nonempty set R together with two binary operations + and . of addition and multiplcation on R for which the following conditions are satisfied :
1) a + b = b + a, 2) (a +' b) + C = a + (b + c), 3) there exists an element O in R such that i; + O a for every a E R, 4) for each a E R, there exists an element -a E R such that a + (-a) = O, 5) (a'b)'c a'(b'c), and 6) a'(b + c) = a'b + a'c and (b + c)'a b'a + c'a,
where it is understood that a, b, c represent arbitrary elements of R. A ring (R, +, .) is saidto be a finite ring if, naturally enough, the set R
of its elements is a finite set. By placing restrictions oI the multiplication operation, several other specialized types of rings are obtained.
Definition 1-2. 1) A commutative ring is a ring (R, +,.) in which multiplication is a commutative operation < a' b = b a for all a, b E R. (In case a'b = b'a for a particular pair a,b, we express this fact by saying that a and b commute.) 2) A ring wth identity is a ring (R, +, .) in which there exists an identity element for the operation of multiplication, normally represented by the symboll, so that a'l l'a = a for aH a E R.
Given a ring (R, +, .) with identity 1, an element a E R is said to be invertible, or to be a unit, whenever a possesses a (two-sided) inv,er'se with respect to multiplication. The multiplicative inverse of a is uniqe, when-ever it exists, and will be denoted by a-l, so thata'a- l = a-l'a t'ln the future, the set of all invertible elements of the ring will be designatel by the symbol R*. It follo~s easily that the system (R*, .) forms a group~ hown as the group 01 invertible elements. In this connection, notice tbat R* is certainly nonempty, for, ifnothing else, 1 and -l'belong to R*. (Qne must not assume, however, that 1 and -1 are necessarily distinct.) ,;'
A consideration of several concrete examples will serve to brii:J.g these ideas into focus.
Example 1-1. If Z, Q, R# denote the sets of integers, rational, and real numbers, respectively, then the systems
(Z,+,), (Q,+,'), (R#,+,') are all examples of rings (here, + and . are taken to be ordinary addition and multiplication). In each oC these cases, the ring is commutative and has the integer 1 for an identity elemento
/
INTRODUCTOR Y CONCEPTS 3
Example 1-2 Let X be a. given set and P(X) be the collection oC all subsets of X. The symmetric difference of two subsets A, B ~ X is the set A I:l B, where
A I:l B =. (A B) u (B - A). If we define addition and multiplication in P(X) by
A + B A I:l B, A B = A n B, then the system (P(X), +, .) forros a commutative ring with identity. The empty set0 serves as the zero element, whereas the multiplicative identity is X. Furthermore, each set in P(X) is its own additive in verse. It is interesting to note that if X is nonempty, then neither (P(X), u, n) nor (P(X), n, u) constitutes a ringo . Example 1-3. Given a ring (R, +, '), we may consider the set M,,(R) oC n x n matrices over R. If 1" {1,2, .. " n}, a typical member oC M,,(R) is a function 1: In X 1" --, R. In practice, one identifies such a Cunction with its values aij I(,}), which are displayed as the n x n rectangular array
( ~: 11 ... ~: 1" ) (aij E R). Il"l , .. a""
For the sake oC simplicity, let us hereafter abbreviate the n x n matrix whose (i,}) entry is aij ~o (a), .
The operations required to make {Mn(R), +, ,) a ring are provlded by the familiar forrpulas
where "
cj = L aik'b'j' k=l i
(We sha11 oCten indulge in this harmless duplcation oC symbols whereby + and . are used with two different meanings.) The zero element oC the resulting ring is the n x n matrix all .of whose entries are O; and -(aij) (-aij)' The ring (Mn(R), +, .) fails to be commutative Cor n > 1.
It is equally easy to show that if (R, +,.) has an identity element 1, then the matrix with l's down the main diagonal (that is, aH = 1) and O's elsewhere will act as identity Cor matrix multiplication. In terms of the Kronecker delta symbol Oij' which is defined by
J1 iC i = j oij = 1,0 ifi =1= j (i,j = 1,2, ... , n),
the identity matrix can be written concisely as (oij)'
2 FIRST COURSE IN RINGS AND IDEALS
can also be otntted when writing out sums and products of more than two elements.
With these remarks in mind, we can now give a more elaborate definition of a ringo A ring (R, +, . ) consists of a nonempty set R together with two binary operations + and . of addition and multiplcation on R for which the following conditions are satisfied :
1) a + b = b + a, 2) (a +' b) + C = a + (b + c), 3) there exists an element O in R such that i; + O a for every a E R, 4) for each a E R, there exists an element -a E R such that a + (-a) = O, 5) (a'b)'c a'(b'c), and 6) a'(b + c) = a'b + a'c and (b + c)'a b'a + c'a,
where it is understood that a, b, c represent arbitrary elements of R. A ring (R, +, .) is saidto be a finite ring if, naturally enough, the set R
of its elements is a finite set. By placing restrictions oI the multiplication operation, several other specialized types of rings are obtained.
Definition 1-2. 1) A commutative ring is a ring (R, +,.) in which multiplication is a commutative operation < a' b = b a for all a, b E R. (In case a'b = b'a for a particular pair a,b, we express this fact by saying that a and b commute.) 2) A ring wth identity is a ring (R, +, .) in which there exists an identity element for the operation of multiplication, normally represented by the symboll, so that a'l l'a = a for aH a E R.
Given a ring (R, +, .) with identity 1, an element a E R is said to be invertible, or to be a unit, whenever a possesses a (two-sided) inv,er'se with respect to multiplication. The multiplicative inverse of a is uniqe, when-ever it exists, and will be denoted by a-l, so thata'a- l = a-l'a t'ln the future, the set of all invertible elements of the ring will be designatel by the symbol R*. It follo~s easily that the system (R*, .) forms a group~ hown as the group 01 invertible elements. In this connection, notice tbat R* is certainly nonempty, for, ifnothing else, 1 and -l'belong to R*. (Qne must not assume, however, that 1 and -1 are necessarily distinct.) ,;'
A consideration of several concrete examples will serve to brii:J.g these ideas into focus.
Example 1-1. If Z, Q, R# denote the sets of integers, rational, and real numbers, respectively, then the systems
(Z,+,), (Q,+,'), (R#,+,') are all examples of rings (here, + and . are taken to be ordinary addition and multiplication). In each oC these cases, the ring is commutative and has the integer 1 for an identity elemento
/
INTRODUCTOR Y CONCEPTS 3
Example 1-2 Let X be a. given set and P(X) be the collection oC all subsets of X. The symmetric difference of two subsets A, B ~ X is the set A I:l B, where
A I:l B =. (A B) u (B - A). If we define addition and multiplication in P(X) by
A + B A I:l B, A B = A n B, then the system (P(X), +, .) forros a commutative ring with identity. The empty set0 serves as the zero element, whereas the multiplicative identity is X. Furthermore, each set in P(X) is its own additive in verse. It is interesting to note that if X is nonempty, then neither (P(X), u, n) nor (P(X), n, u) constitutes a ringo . Example 1-3. Given a ring (R, +, '), we may consider the set M,,(R) oC n x n matrices over R. If 1" {1,2, .. " n}, a typical member oC M,,(R) is a function 1: In X 1" --, R. In practice, one identifies such a Cunction with its values aij I(,}), which are displayed as the n x n rectangular array
( ~: 11 ... ~: 1" ) (aij E R). Il"l , .. a""
For the sake oC simplicity, let us hereafter abbreviate the n x n matrix whose (i,}) entry is aij ~o (a), .
The operations required to make {Mn(R), +, ,) a ring are provlded by the familiar forrpulas
where "
cj = L aik'b'j' k=l i
(We sha11 oCten indulge in this harmless duplcation oC symbols whereby + and . are used with two different meanings.) The zero element oC the resulting ring is the n x n matrix all .of whose entries are O; and -(aij) (-aij)' The ring (Mn(R), +, .) fails to be commutative Cor n > 1.
It is equally easy to show that if (R, +,.) has an identity element 1, then the matrix with l's down the main diagonal (that is, aH = 1) and O's elsewhere will act as identity Cor matrix multiplication. In terms of the Kronecker delta symbol Oij' which is defined by
J1 iC i = j oij = 1,0 ifi =1= j (i,j = 1,2, ... , n),
the identity matrix can be written concisely as (oij)'
4 FIRST COURSE IN RINGS AND IDEALS , ,
Example 1-4. To develop our next example, let X be an arbitrary (non- ' enipty) set and(R, +, .) be a ringo We adoptthe notation map(X, R} for the set consisting f a11 mappings from ,X into R; in symbols,
map(X, R) :: {JI!: X ~ R}. (Foreaseofnotation, let usalso agree to write map R in place o~map(R, R):) Now, the elements ofmap(X, R) can be combined by performmg algebralc operatio:ps ontheir functional values. More specifically, the poiritwise sum and producto'r f and g, denoted by f + g, and f g,. respectively, are the functions whih satisfy
, (f + g)(X) = f(x) + g(x), (f'g)(x) = f(x)'g(x),(x E X). It is read~1i'~erified thaJ the aboye definitions provide map(X, R) with
the structuriola ringo We simply point out that tbe zero element of tbis ring is the c9~stant functin wbose sole value is 0, and the additive inverse -foffis cha,racterized by the rule (-1)(x) = -f(x).
Notice that,'the aIgebraic properties of map(X, R) are determined by what happens in thering (R, +,.) (the set X fumishes only the points for the pointwise operations).' , For instance, if (R, +, .) has a multipli.cative identity 1, then the ring (map(X, R), +,.) likewise possesses an identity element; namely, the constant functit;>n defined by l(x) = 1 f6r all x E. X. ' Example 1-5 .. Our final example i8 that of the ring oC integers modulo n, wbere n is a fixed positive integer. In order to describe tbis system, we first introduce th notion of congruence: two integers a and b are said to be eongruent modulo, n, written a == b (mod n), if and only if the difference a b is divisible by n; in other words, a == b (mod n) if nd only if a - b kn for some k E Z. We leave the reader to convince himself that the relation "congruent modulo n" defines an equivalence relation on the
, set Z of integers. As such, it partitions Z into disjoint c1asses of congruent e1ements, caBed eongruenee classes. For each integer a, let the congnence class to which a belongs be denoted by [a J:
[aJ = {xe Zlx E a (mod n)} = {a + knlkeZ}.
Of course, the same congrtience class may very well arise from another integer; any integer a' Cor which [a/J = [a J is said to be a representative of [a]. Qne final, purely notational, remark : the collection of a11 congruence classes oCintegers modulo n will be designated by Zn'
It can be shown tbat the congruence cIasses [OJ, [lJ, ... , [n - 1J exhaust the elements of Z.. Given 'an arbitrary integer a, the division algorithm asserts tbat there exist uniqueq, re Z, with O :s;; r < n, such that a = qn + r. By the definition of congruence, a == r (rnod n), or
I t ,
,
,1 {'
. \.
INTRODUCTORY CONCEPTS 5
equivalently, [a] = [rJ. Thus, there are' at most n different congruence . claSses in Zn, namely, [OJ, [lJ, ... , [n - 1]. But these n c1assesare them-
'selvesdistinct. For ifO :s;; b < a < n, then O < a - b < n and so a - b cannot be divisible by n, whence [a J =1 [b]. Accordingly, Zn cqnsists of exactly n eIements:
Zn = {[OJ, [lJ, ... , [n - 1J}. Thert::ader should J:eep in nlind that each congruence class li~ted.,above is determined by any one of its members; a1l we have done is to f~present the cIass by its smallest nonnegative representative. .':'
Our next step i5 to define the manner in which the members,ol Zn are to be added and multiplied, so tbat the.resulting system will forma ringo Tbe definitions are as follows: for each [a J, [b J E Zn, ,, "
[aJ +n [bJ [a + bJ, . [ab]. : .... fr.
In other words, the sum and product of two congruence c1asses[li'nd [b J . are the unique members of Zn which contain' the ordinary sumo a. eV' b and ordinary product ab;respectively. Before considering the algebraic properties ofthese operations, it is necessary to make certain tbat they are well-defined' and do not depend upon which representatives of the congruence c1asses are chosen. In regaro to multiplication, for instance, we want to satisfy ourselves that if [a'J= [aJ and [b'J = [bJ, then [a'h[b'J = [aln[bJ, or, .rather, that [a'b'J == [ah]. Now, al E [a'J = [aJ and b' e [b'] [bJ, which signifies that a' = a' + kn and bl = b + jn for some k, j e Z. But then
a'b l = (a + kn)(b + jn) ab + (aj + bk + kjn)n . Hence, a'b' == ab (mod n) and so [a'b'] = [ab J, as desired. The proof that addition is unambiguously defined proceeds similarIy.
We omit the detailed verification of the faet tbat (Z., +n' 'n) is a com-mutatve ring with identity (tradtioIiaIly known as the ring of integers moaulo n), remarking only that the various ring axiom8 hold in Zn simply because they hold in Z. The distributive law, for instance, fo11ows in Zn from its validity in Z: '
[aL([bJ +" [eJ) = [aL[b + eJ = [a(b + e)] [ab + aeJ = [abJ +n [aeJ
= [aL[bJ +11 [aL [e]. Notice, too, that the congruence classes [OJ and [lJ serve as the zero element and multiplieative identity, respectively, whereas [-aJ i8 the additive in verse of [aJ in Zn' When no eonfusion is likely, we sha1l1eave off the brackets from the elements of Zn, thereby making no genuine distnctiori
4 FIRST COURSE IN RINGS AND IDEALS , ,
Example 1-4. To develop our next example, let X be an arbitrary (non- ' enipty) set and(R, +, .) be a ringo We adoptthe notation map(X, R} for the set consisting f a11 mappings from ,X into R; in symbols,
map(X, R) :: {JI!: X ~ R}. (Foreaseofnotation, let usalso agree to write map R in place o~map(R, R):) Now, the elements ofmap(X, R) can be combined by performmg algebralc operatio:ps ontheir functional values. More specifically, the poiritwise sum and producto'r f and g, denoted by f + g, and f g,. respectively, are the functions whih satisfy
, (f + g)(X) = f(x) + g(x), (f'g)(x) = f(x)'g(x),(x E X). It is read~1i'~erified thaJ the aboye definitions provide map(X, R) with
the structuriola ringo We simply point out that tbe zero element of tbis ring is the c9~stant functin wbose sole value is 0, and the additive inverse -foffis cha,racterized by the rule (-1)(x) = -f(x).
Notice that,'the aIgebraic properties of map(X, R) are determined by what happens in thering (R, +,.) (the set X fumishes only the points for the pointwise operations).' , For instance, if (R, +, .) has a multipli.cative identity 1, then the ring (map(X, R), +,.) likewise possesses an identity element; namely, the constant functit;>n defined by l(x) = 1 f6r all x E. X. ' Example 1-5 .. Our final example i8 that of the ring oC integers modulo n, wbere n is a fixed positive integer. In order to describe tbis system, we first introduce th notion of congruence: two integers a and b are said to be eongruent modulo, n, written a == b (mod n), if and only if the difference a b is divisible by n; in other words, a == b (mod n) if nd only if a - b kn for some k E Z. We leave the reader to convince himself that the relation "congruent modulo n" defines an equivalence relation on the
, set Z of integers. As such, it partitions Z into disjoint c1asses of congruent e1ements, caBed eongruenee classes. For each integer a, let the congnence class to which a belongs be denoted by [a J:
[aJ = {xe Zlx E a (mod n)} = {a + knlkeZ}.
Of course, the same congrtience class may very well arise from another integer; any integer a' Cor which [a/J = [a J is said to be a representative of [a]. Qne final, purely notational, remark : the collection of a11 congruence classes oCintegers modulo n will be designated by Zn'
It can be shown tbat the congruence cIasses [OJ, [lJ, ... , [n - 1J exhaust the elements of Z.. Given 'an arbitrary integer a, the division algorithm asserts tbat there exist uniqueq, re Z, with O :s;; r < n, such that a = qn + r. By the definition of congruence, a == r (rnod n), or
I t ,
,
,1 {'
. \.
INTRODUCTORY CONCEPTS 5
equivalently, [a] = [rJ. Thus, there are' at most n different congruence . claSses in Zn, namely, [OJ, [lJ, ... , [n - 1]. But these n c1assesare them-
'selvesdistinct. For ifO :s;; b < a < n, then O < a - b < n and so a - b cannot be divisible by n, whence [a J =1 [b]. Accordingly, Zn cqnsists of exactly n eIements:
Zn = {[OJ, [lJ, ... , [n - 1J}. Thert::ader should J:eep in nlind that each congruence class li~ted.,above is determined by any one of its members; a1l we have done is to f~present the cIass by its smallest nonnegative representative. .':'
Our next step i5 to define the manner in which the members,ol Zn are to be added and multiplied, so tbat the.resulting system will forma ringo Tbe definitions are as follows: for each [a J, [b J E Zn, ,, "
[aJ +n [bJ [a + bJ, . [ab]. : .... fr.
In other words, the sum and product of two congruence c1asses[li'nd [b J . are the unique members of Zn which contain' the ordinary sumo a. eV' b and ordinary product ab;respectively. Before considering the algebraic properties ofthese operations, it is necessary to make certain tbat they are well-defined' and do not depend upon which representatives of the congruence c1asses are chosen. In regaro to multiplication, for instance, we want to satisfy ourselves that if [a'J= [aJ and [b'J = [bJ, then [a'h[b'J = [aln[bJ, or, .rather, that [a'b'J == [ah]. Now, al E [a'J = [aJ and b' e [b'] [bJ, which signifies that a' = a' + kn and bl = b + jn for some k, j e Z. But then
a'b l = (a + kn)(b + jn) ab + (aj + bk + kjn)n . Hence, a'b' == ab (mod n) and so [a'b'] = [ab J, as desired. The proof that addition is unambiguously defined proceeds similarIy.
We omit the detailed verification of the faet tbat (Z., +n' 'n) is a com-mutatve ring with identity (tradtioIiaIly known as the ring of integers moaulo n), remarking only that the various ring axiom8 hold in Zn simply because they hold in Z. The distributive law, for instance, fo11ows in Zn from its validity in Z: '
[aL([bJ +" [eJ) = [aL[b + eJ = [a(b + e)] [ab + aeJ = [abJ +n [aeJ
= [aL[bJ +11 [aL [e]. Notice, too, that the congruence classes [OJ and [lJ serve as the zero element and multiplieative identity, respectively, whereas [-aJ i8 the additive in verse of [aJ in Zn' When no eonfusion is likely, we sha1l1eave off the brackets from the elements of Zn, thereby making no genuine distnctiori
6 FIRST COURSE IN RINGS AND IDEALS
betwec3D a congruence c1ass and its smallest nnnegative representative; under this convention, Z" = {O, 1, ... , n - 1}. It is perhaps worth com-menting that, since Z 1 = Z, a number of texts specifically exc1ude the value 1 for n.
Although it is logically correct (and often convenient) to speak of a ring as an ordered triple, the notation becomes unwieldy as one progresses further into the theory. We shall therefore adopt the usual convention of designating a ring, say (R, +, '), simply by the set symbol R and assume that + and . are known. The reader should realize, however, that a given set may perfectly well be the underlying set of several different rings. Let us also agree to abbreviate a + (-b) as a - b and subsequently refer to tbis expression as the difference between a and b. As a final concession to brevity, juxtaposition without a median dot will be used to denote the product oftwo ring elements. .
With these conventions on record, let us begin our formal development of ring theory. The material covered in the next several pages will probably be familiar to most readers and is inc1uded more to assure completeness than to present new ideas.
Theorem 1-1. If R is a ring, then for any a, b, e E R 1) Oa = aO = O, 2) a(-b) = (-a)b = -(ab), 3) (-a)(-b) = ab, and 4) a(b - e) = ab - ae, (b - e)a = ba - ea.
Proof. These turn out, in,t.he m~in, to be simple consequences of the dis-tributive laws. For instari2e, Irom O + O = O, it follows that
Oa=;(O + O)a = Oa + Da. Thus, by the cancellation law:ror the additive group (R, +), we have Oa = O. In a like manner, one obtains aO = O. The proof of (2) requires the fact that each element of R h~SI a unique additive inverse (Problem 1). Since b + (-b) = O, '"
ab + a("':'~) = a(b + (-b)) = aO = O, which then implies that -(ab) = a( -b). The argument that (-a)b is also the additive inverse of ab proceeds similarly. Tbis leads immediately'to (3):
(-a)( -b) = -( -a)b = - (-(ab)) = abo The last assertion is all but obvious.
. There is one very simple ring that consists only of the additive identity O, with addition and multiplication given by O + O = O, 00 = O; tbis ring is usually called the trivial ringo '
INTRODUCTOR Y CONCEPTS 7
Corollary. Let R be a ring with identity 1. If R is noUhe trivial ring, then the elements O and 1 are distinct.
Proof. Since R =1= {O}, there exists sorne nonzero element a E R. If O and 1 were equal, it would follow that a = al .. = aO = O, an obvious contradic-tion.
CONVENTION: Let us assume, once and for all, that any ring with identity contains more than oneelement. This will rule out the possibility that O and 1 coincide.
We now make several remarksabout the concept of zero divisors (the term "divisors of zero" is also in common use): .
Defution 1-3. If R is a ring and O =1= a E R, then a is called a left (right) zero divisor in R if there exists sorne b =1= O in R such that ab = O (ba = O). A zero divisor is any element of R that is either a left or right zero divisor.
According to this definition, O is not a zero divisor, and if R contains an identity 1, then 1 is not a zero divisor nor is any element of R which happens to possess a multiplicative inverse. An obvious example of a riIig with zero divisors is Z., where the integer n > 1 is composite; if n = n1n2 in Z (O < n1, n2 < n), then the product n1 .n2 = O in Z .
For the most part, we shall be studying rings without zero divisors. In such rings it is possible to conc1ude from the equation ab = O that either a = O or b = O.
One can express the property of being with or without zero divisors in the following useful way.
,Theorem 1-2. A ring R is without zero divisors if and only if it satisfies the cancellation laws for multiplication; that is, for all a, b, e E R, ab = ae and ba = ca, where a =1= O, implies b = e.
Proof. Suppose that R is without zero divisors and let ab = ae, a =1= O. Then, the product a(b - e) = O, which means that b - e = O and b = e. The argument is the same for the equation ba = ca. Conversely, let R satisfy the cancellation laws and assume that ab = O, with a =1= O. We then' have ab = aO, whence by cancellation b = O. Similarly, b =1= O implies a = O, proving that there are no zero divisors in R. '
By an integral domain is meant a commutative ring with identity which has no zero divisors, Perhaps the best-known example ofan integral domain is the ring ofintegers; hence the choice ofterminology. Theorem 1-2 shows that the cancellation laws for multiplication hold in any integral domain.
The reader should be warned that many authors do not insist on the presence of a multiplicative identity when defining integral domains; and
6 FIRST COURSE IN RINGS AND IDEALS
betwec3D a congruence c1ass and its smallest nnnegative representative; under this convention, Z" = {O, 1, ... , n - 1}. It is perhaps worth com-menting that, since Z 1 = Z, a number of texts specifically exc1ude the value 1 for n.
Although it is logically correct (and often convenient) to speak of a ring as an ordered triple, the notation becomes unwieldy as one progresses further into the theory. We shall therefore adopt the usual convention of designating a ring, say (R, +, '), simply by the set symbol R and assume that + and . are known. The reader should realize, however, that a given set may perfectly well be the underlying set of several different rings. Let us also agree to abbreviate a + (-b) as a - b and subsequently refer to tbis expression as the difference between a and b. As a final concession to brevity, juxtaposition without a median dot will be used to denote the product oftwo ring elements. .
With these conventions on record, let us begin our formal development of ring theory. The material covered in the next several pages will probably be familiar to most readers and is inc1uded more to assure completeness than to present new ideas.
Theorem 1-1. If R is a ring, then for any a, b, e E R 1) Oa = aO = O, 2) a(-b) = (-a)b = -(ab), 3) (-a)(-b) = ab, and 4) a(b - e) = ab - ae, (b - e)a = ba - ea.
Proof. These turn out, in,t.he m~in, to be simple consequences of the dis-tributive laws. For instari2e, Irom O + O = O, it follows that
Oa=;(O + O)a = Oa + Da. Thus, by the cancellation law:ror the additive group (R, +), we have Oa = O. In a like manner, one obtains aO = O. The proof of (2) requires the fact that each element of R h~SI a unique additive inverse (Problem 1). Since b + (-b) = O, '"
ab + a("':'~) = a(b + (-b)) = aO = O, which then implies that -(ab) = a( -b). The argument that (-a)b is also the additive inverse of ab proceeds similarly. Tbis leads immediately'to (3):
(-a)( -b) = -( -a)b = - (-(ab)) = abo The last assertion is all but obvious.
. There is one very simple ring that consists only of the additive identity O, with addition and multiplication given by O + O = O, 00 = O; tbis ring is usually called the trivial ringo '
INTRODUCTOR Y CONCEPTS 7
Corollary. Let R be a ring with identity 1. If R is noUhe trivial ring, then the elements O and 1 are distinct.
Proof. Since R =1= {O}, there exists sorne nonzero element a E R. If O and 1 were equal, it would follow that a = al .. = aO = O, an obvious contradic-tion.
CONVENTION: Let us assume, once and for all, that any ring with identity contains more than oneelement. This will rule out the possibility that O and 1 coincide.
We now make several remarksabout the concept of zero divisors (the term "divisors of zero" is also in common use): .
Defution 1-3. If R is a ring and O =1= a E R, then a is called a left (right) zero divisor in R if there exists sorne b =1= O in R such that ab = O (ba = O). A zero divisor is any element of R that is either a left or right zero divisor.
According to this definition, O is not a zero divisor, and if R contains an identity 1, then 1 is not a zero divisor nor is any element of R which happens to possess a multiplicative inverse. An obvious example of a riIig with zero divisors is Z., where the integer n > 1 is composite; if n = n1n2 in Z (O < n1, n2 < n), then the product n1 .n2 = O in Z .
For the most part, we shall be studying rings without zero divisors. In such rings it is possible to conc1ude from the equation ab = O that either a = O or b = O.
One can express the property of being with or without zero divisors in the following useful way.
,Theorem 1-2. A ring R is without zero divisors if and only if it satisfies the cancellation laws for multiplication; that is, for all a, b, e E R, ab = ae and ba = ca, where a =1= O, implies b = e.
Proof. Suppose that R is without zero divisors and let ab = ae, a =1= O. Then, the product a(b - e) = O, which means that b - e = O and b = e. The argument is the same for the equation ba = ca. Conversely, let R satisfy the cancellation laws and assume that ab = O, with a =1= O. We then' have ab = aO, whence by cancellation b = O. Similarly, b =1= O implies a = O, proving that there are no zero divisors in R. '
By an integral domain is meant a commutative ring with identity which has no zero divisors, Perhaps the best-known example ofan integral domain is the ring ofintegers; hence the choice ofterminology. Theorem 1-2 shows that the cancellation laws for multiplication hold in any integral domain.
The reader should be warned that many authors do not insist on the presence of a multiplicative identity when defining integral domains; and
8. FIRST COURSE IN RINGS AND IDEALS
in this case the term "integral domain" would merely indicate a commutative ring without zeto divisors.
We change direction somewhat to deal with the situation where a subset of a ring again constitutes a ringo Formally speaking,
Definition 1-4. Let (R, +, .) be a ring and S 5; R be a nonempty subset of R. Ifthe system (S, +, .) is itselfa ring (using the induced operations), then (S, +, .) is said to be a subring of (R, +, '). This o definition is adequate, but unwieldy, siIice all the aspects of the
definition of a ring must be checked in deteimining whether a given subset is a subring. In seeking a simpler criterion, noticethat (S, +, .) is a subring of (R, +, .) provided that (S, +) is a subgroup of (R, +), (S, .) is a subsemi-: group of (R, '), and the two distributive laws are .satisfied in S. But the distributive and associative laws hold automaticaJly for elements of S as a consequence of their validity in R.
o
Since these laws are inherited from R, there is no necessity of requiring them in the definition of a subring.
Taking our cuefrom these remarks, a subring could just as well be defined as follows. The system (S, +, o) forms a subring of the ring (R, +, .) if and only if .
1) S is a nonempty subset of R, 2) (S, +) is a subgroup of(R, +), and 3) the set S is closed under multiplication.
To add th~ final touch, even this definition can be improved upon; for the reader versed in group theory will recall that (S, +) is a subgroup of the group (R, +) provided that a - b E S whenever a, bES. By these observations we are led toa set of c10sure conditions wruch make it some-what easier to verify tbat a particular subset is actually a subring.
Theorem 1-3. Let R be a ring and 0 =1= S 5; R. Theri, S i8 a subring of R if and only if 1) a, b E S imply a - b E S 2) a, b E S imply ab E S
. (closure under differences), (closure under multplication).
If S is a subring of the ring R, then the zero element of S is that of R and, moreover, the additive inverse of an element of the subring S is the same as its inverse as a member of R, Verification of these assertions is left as an exercise.
Example 1-6. Every ring R has two obvious subrings, namely, the set {O}, consisting only of the zero element, and R itself. These two subrings are usually referred to as the trivial subrings of R; all other subrings (if any exist) are called nontrivial. We shall use the term proper subring to mean a subring which is different from R.
INTRODUCTORY CONCEPTS 9
Example 1-7. The set.Z. of ev~n integers forms a subring of the ring Z of integers, for
2n -:- 2m = 2(n - m) E Z., o (2n) (2m) = 2(2nm) E Z .
This example al so illustrates a fact worth bearing in mind: in a ring with identity, a subring need nof contain the identity elemento
Prior to stating our next theorem, let us define the center of a ring R, denoted by cent R, to be th set
1'"
cent R=' {a E RJar = ra for all r ER}. Phrased otherwise, cenot R;:consists of those elements which conimute with every member of R. It s~6uld be apparent that a ring R is commutative if and only if cent R = R."::::.
',.',
Theoreinl-4. For ahy.ring R, cent R is a subring of R..~~I' Proof. To be conscientiouiabout details,first observe that 'tentRis non-empty; for, at the very least, the zero element O E R. Now pick any two elements a, b in cent R. By the definition of center, we know that ar = ra and br :;= rb for every choice ofr E R. Thus, for arbitrary rE R,
(a - b)r = ar - br = ra - rb = r(a - b), which implies that a - b E cent R. A similar argumentaffirms that the product ab also lies in cent R. In the lightof Theorem 1-3, theseare sufficient conditions for the centet to be a subring of R.
It has aIread y been remarked that, when a ring has an identity, this need not be true of its subrings. Other interesting situations may arise,
1) Sorne subfing has a multiplitative identity, but the entire ring does noto 2) Boththe ring and one ofits subrings possess identity elements, but they
are distinct. .
In each of the cited cases the identity for the subring is necessarily a divisor of zero in the larger ringo To justify this claim, let l' =1= O denote the identity element of the subririg S; we assume further that l' does not act as an identity for the whole ring R. Accordingly, there exists sorne element a E R for which al'. =1= a. It is dear that
(al')l' = a(l'l') = al', or (al' - a)l' = O. Since rieither al' -' a nor l' is zero, the ring R has zero divisors, and in particular l' is a zerp divisor.
Example 1-8. to present a simple illustration of a ring in which the seoond . of the aforementioned possibilities occurs, consider the set R = Z x Z,
8. FIRST COURSE IN RINGS AND IDEALS
in this case the term "integral domain" would merely indicate a commutative ring without zeto divisors.
We change direction somewhat to deal with the situation where a subset of a ring again constitutes a ringo Formally speaking,
Definition 1-4. Let (R, +, .) be a ring and S 5; R be a nonempty subset of R. Ifthe system (S, +, .) is itselfa ring (using the induced operations), then (S, +, .) is said to be a subring of (R, +, '). This o definition is adequate, but unwieldy, siIice all the aspects of the
definition of a ring must be checked in deteimining whether a given subset is a subring. In seeking a simpler criterion, noticethat (S, +, .) is a subring of (R, +, .) provided that (S, +) is a subgroup of (R, +), (S, .) is a subsemi-: group of (R, '), and the two distributive laws are .satisfied in S. But the distributive and associative laws hold automaticaJly for elements of S as a consequence of their validity in R.
o
Since these laws are inherited from R, there is no necessity of requiring them in the definition of a subring.
Taking our cuefrom these remarks, a subring could just as well be defined as follows. The system (S, +, o) forms a subring of the ring (R, +, .) if and only if .
1) S is a nonempty subset of R, 2) (S, +) is a subgroup of(R, +), and 3) the set S is closed under multiplication.
To add th~ final touch, even this definition can be improved upon; for the reader versed in group theory will recall that (S, +) is a subgroup of the group (R, +) provided that a - b E S whenever a, bES. By these observations we are led toa set of c10sure conditions wruch make it some-what easier to verify tbat a particular subset is actually a subring.
Theorem 1-3. Let R be a ring and 0 =1= S 5; R. Theri, S i8 a subring of R if and only if 1) a, b E S imply a - b E S 2) a, b E S imply ab E S
. (closure under differences), (closure under multplication).
If S is a subring of the ring R, then the zero element of S is that of R and, moreover, the additive inverse of an element of the subring S is the same as its inverse as a member of R, Verification of these assertions is left as an exercise.
Example 1-6. Every ring R has two obvious subrings, namely, the set {O}, consisting only of the zero element, and R itself. These two subrings are usually referred to as the trivial subrings of R; all other subrings (if any exist) are called nontrivial. We shall use the term proper subring to mean a subring which is different from R.
INTRODUCTORY CONCEPTS 9
Example 1-7. The set.Z. of ev~n integers forms a subring of the ring Z of integers, for
2n -:- 2m = 2(n - m) E Z., o (2n) (2m) = 2(2nm) E Z .
This example al so illustrates a fact worth bearing in mind: in a ring with identity, a subring need nof contain the identity elemento
Prior to stating our next theorem, let us define the center of a ring R, denoted by cent R, to be th set
1'"
cent R=' {a E RJar = ra for all r ER}. Phrased otherwise, cenot R;:consists of those elements which conimute with every member of R. It s~6uld be apparent that a ring R is commutative if and only if cent R = R."::::.
',.',
Theoreinl-4. For ahy.ring R, cent R is a subring of R..~~I' Proof. To be conscientiouiabout details,first observe that 'tentRis non-empty; for, at the very least, the zero element O E R. Now pick any two elements a, b in cent R. By the definition of center, we know that ar = ra and br :;= rb for every choice ofr E R. Thus, for arbitrary rE R,
(a - b)r = ar - br = ra - rb = r(a - b), which implies that a - b E cent R. A similar argumentaffirms that the product ab also lies in cent R. In the lightof Theorem 1-3, theseare sufficient conditions for the centet to be a subring of R.
It has aIread y been remarked that, when a ring has an identity, this need not be true of its subrings. Other interesting situations may arise,
1) Sorne subfing has a multiplitative identity, but the entire ring does noto 2) Boththe ring and one ofits subrings possess identity elements, but they
are distinct. .
In each of the cited cases the identity for the subring is necessarily a divisor of zero in the larger ringo To justify this claim, let l' =1= O denote the identity element of the subririg S; we assume further that l' does not act as an identity for the whole ring R. Accordingly, there exists sorne element a E R for which al'. =1= a. It is dear that
(al')l' = a(l'l') = al', or (al' - a)l' = O. Since rieither al' -' a nor l' is zero, the ring R has zero divisors, and in particular l' is a zerp divisor.
Example 1-8. to present a simple illustration of a ring in which the seoond . of the aforementioned possibilities occurs, consider the set R = Z x Z,
10 FIRST COURSE IN RINGS ANO IDEALS
consisting of ordered pairs of integers. One converts R into a ring by defining addition and rnultiplicatiori componentwise :
(a, b) + (e, d) = (a + e, b + d), (a,b)(e, d) = (ae, bd).
A routne calculation will show that Z x {O} = {(a, O)la E Z} forms a sub-ring with .identty element (1, O). This obviously differs from the identity of the entire ring R, which turns out to be the ordered pair (1, 1). By our prevous rernarks, (1, O)rnustbeazerodivisorinR;infact,(l, 0)(0,1) = (O, O), where (O, O) serves as the zero element of R.
If R is an arbitrary ring and n a positive integer, then the nth power a" of an element a E R is defined by the inductve condtions al = a and a" = a"-la. Frorn tbis the usuallaws of exponents follow at once:
a"am
= an+m, (a"t = a"m (n, m E Z+). To establish these rules, fu m and proceed by induction on n. Observe also that iftwo elements a, bE R happen to cornrnute, so do all powers of a and b, whence (ab)" = a"b" for each positive integer n.
In the event that R possesses an identty element 1 and a- l exists, negave powers of a can be introduced by intelJ)retng a-" as (a- l )", where n > O. With the definition aO = 1, the symbol a" now has a well-defined meaning for every integer n (at least when attention is restricted to invertible elernents).
Parallelng the exponent notation for powers, there is the notation of integral multiples of an e1ement a E R. For each positive integer n, we define the nth natural multiple na recursively as follows:
la = a and na = (n - l)a + a, when n > 1. If t is also agreed to let Oa = O and ( - n)a = - (na), then the definition of na can be extended to all ntegers. Integral multiples satisfy several identities which are easy to establish:
(n + m)a = na + ma, (nm)a = n(ma),
n(a + b) = /la + nb, for a, b E R and arbitrary integers n and m. In addition to these rules, there are two further properties resulting frorn the distributive law, namely,
n(ab) = (na)b = a(nb), and (na)(mb) = (nm)(ab). Experience impels us to emphasize that the expression na should not
be regarded as a ring product (indeed, the integer n 'may not even be a member of R); the entire symbol na is just a convenient way of indicating
INTROOUCTORY CONCEPTS 11
a certain sum of elements of R. However, when there is an identity for rnultiplication, it is possible to represent na as a product oftwo ring elements, namely, na = (n1)a.
To proceed further with our lmited program, we must first frame a defintion.
Definition 1-5. Let R be an arbtrary ringo If there exists a postive integer n such that na = O for all a E R, then the srnaIlest positive integer wth this property is called the eharaeteristie of the ringo If no such positive integer exists (that is, n = O is the only integer for which na =. O for all a in R), then R is said to be of eharaeteristie zero. We shall wnte char R for the characteristc of R.
The rings of integers, rational numbers, and real numbers are all standard exarnples of systerns having characteristc zero (sorne writers prefer the expression "characteristic infinity"). On the other hand, the ring P(X) of subsets of a fixed set X is of characteristic 2, since
2A = A A A = (A - A) u (A - A) = rp for every subset A S; X.
Although the definition of characteristc makes an assertion about every element of the ring, in rings with identty the characteristic is completely determined by the identity elernent. . We reach tbis conc1usion below.
Theorem 1-5. If R is any ring wth identity 1, then R has characteristic n > O if and only if n is the least postive integer for which nI = O.
Proo/. If char R = n > O, then na = O for every a E R and so, in particular, nI = O. Were ml = O, where O < m < n, jt wciuld necessarily follow that
ma = m(la) = (m1)a = Oa = O for every elernent a E R. The implcaton is that char R < n, which is impossible. One establishes the converse in rnuch the 'sarne way.
As we have seen, multiplcation exerts a strong infiuence on the addtive structure of a ring through the distributive law. The following corollary to Theorem 1-5 shows that by sufficiently restricting the multiplcation in a ring R it is possible to reach sorne interestng conc1usions regarding the characteristic of R.
Corollary 1. In an integral dornain R all the nonzero elernents have the sarne addtive order; this order is the characteristic of the domain when char R > O and infinite when char R = o.
Proo/. To verify this assertion, suppose first that char R = n > O. Accord-ing to the defintion of characteristic, each element O =1= a E R will then possess a finte additive order m, wth m :::;; n. (Recall that for an element
10 FIRST COURSE IN RINGS ANO IDEALS
consisting of ordered pairs of integers. One converts R into a ring by defining addition and rnultiplicatiori componentwise :
(a, b) + (e, d) = (a + e, b + d), (a,b)(e, d) = (ae, bd).
A routne calculation will show that Z x {O} = {(a, O)la E Z} forms a sub-ring with .identty element (1, O). This obviously differs from the identity of the entire ring R, which turns out to be the ordered pair (1, 1). By our prevous rernarks, (1, O)rnustbeazerodivisorinR;infact,(l, 0)(0,1) = (O, O), where (O, O) serves as the zero element of R.
If R is an arbitrary ring and n a positive integer, then the nth power a" of an element a E R is defined by the inductve condtions al = a and a" = a"-la. Frorn tbis the usuallaws of exponents follow at once:
a"am
= an+m, (a"t = a"m (n, m E Z+). To establish these rules, fu m and proceed by induction on n. Observe also that iftwo elements a, bE R happen to cornrnute, so do all powers of a and b, whence (ab)" = a"b" for each positive integer n.
In the event that R possesses an identty element 1 and a- l exists, negave powers of a can be introduced by intelJ)retng a-" as (a- l )", where n > O. With the definition aO = 1, the symbol a" now has a well-defined meaning for every integer n (at least when attention is restricted to invertible elernents).
Parallelng the exponent notation for powers, there is the notation of integral multiples of an e1ement a E R. For each positive integer n, we define the nth natural multiple na recursively as follows:
la = a and na = (n - l)a + a, when n > 1. If t is also agreed to let Oa = O and ( - n)a = - (na), then the definition of na can be extended to all ntegers. Integral multiples satisfy several identities which are easy to establish:
(n + m)a = na + ma, (nm)a = n(ma),
n(a + b) = /la + nb, for a, b E R and arbitrary integers n and m. In addition to these rules, there are two further properties resulting frorn the distributive law, namely,
n(ab) = (na)b = a(nb), and (na)(mb) = (nm)(ab). Experience impels us to emphasize that the expression na should not
be regarded as a ring product (indeed, the integer n 'may not even be a member of R); the entire symbol na is just a convenient way of indicating
INTROOUCTORY CONCEPTS 11
a certain sum of elements of R. However, when there is an identity for rnultiplication, it is possible to represent na as a product oftwo ring elements, namely, na = (n1)a.
To proceed further with our lmited program, we must first frame a defintion.
Definition 1-5. Let R be an arbtrary ringo If there exists a postive integer n such that na = O for all a E R, then the srnaIlest positive integer wth this property is called the eharaeteristie of the ringo If no such positive integer exists (that is, n = O is the only integer for which na =. O for all a in R), then R is said to be of eharaeteristie zero. We shall wnte char R for the characteristc of R.
The rings of integers, rational numbers, and real numbers are all standard exarnples of systerns having characteristc zero (sorne writers prefer the expression "characteristic infinity"). On the other hand, the ring P(X) of subsets of a fixed set X is of characteristic 2, since
2A = A A A = (A - A) u (A - A) = rp for every subset A S; X.
Although the definition of characteristc makes an assertion about every element of the ring, in rings with identty the characteristic is completely determined by the identity elernent. . We reach tbis conc1usion below.
Theorem 1-5. If R is any ring wth identity 1, then R has characteristic n > O if and only if n is the least postive integer for which nI = O.
Proo/. If char R = n > O, then na = O for every a E R and so, in particular, nI = O. Were ml = O, where O < m < n, jt wciuld necessarily follow that
ma = m(la) = (m1)a = Oa = O for every elernent a E R. The implcaton is that char R < n, which is impossible. One establishes the converse in rnuch the 'sarne way.
As we have seen, multiplcation exerts a strong infiuence on the addtive structure of a ring through the distributive law. The following corollary to Theorem 1-5 shows that by sufficiently restricting the multiplcation in a ring R it is possible to reach sorne interestng conc1usions regarding the characteristic of R.
Corollary 1. In an integral dornain R all the nonzero elernents have the sarne addtive order; this order is the characteristic of the domain when char R > O and infinite when char R = o.
Proo/. To verify this assertion, suppose first that char R = n > O. Accord-ing to the defintion of characteristic, each element O =1= a E R will then possess a finte additive order m, wth m :::;; n. (Recall that for an element
12 FIRST COURSE IN RINGS AND IDEALS
a + O ofthe group (R, +) to have order m mean s that ma = O and ka + O irO < k < m.) But the retation O == ma = (ml)a mplies that mI = O, Cor R is assumed to be free of zero divisors. We therefore conclude from the theorem that n ::;; m, whence m and n are equal. In consequence, every nonzero element of R has additive order n.
A somewhat similar argumen't can be employed when char R O. The equaqon ma = O would lead, as before, to m1 = O or m O. In this case every nonzero eIement a E R mus be oC infinite order.
The last result seives to bring OUt anotner use fuI point, which we place on record as . .
CoroHary 2. only iC na
An integral dornain R has positive characteristic ifand O foro s?me O + a E R and some integer n E Z+.
Continuing in thls veln, let us next show tbat not any commutative group can serve as the additive group of an integral domain.
Theorem 1-6. The cbaracteristic of an integral domain is either zero or a prime number. '
Prooj. Let R be of positive characteristic n and assume that n is not a prime. Then, n has a nonrivial factorization n = nI n2 , with I < nI' n2 < n. It follows tbat
O = nl = (n 1n2)1 = (n In2)12 = (n I l)(n21). By supposition,.R is without zero divisors, so that either nll O or n2 I O. Since both nI and n2 are less than n, this contradicts the choice of n as the . leastpositive integer for which n1 = O. We therefore concluoe that char R must be prime.
CorolIary .. If R is a finite integral domain, then char R = .p, a prime. Turning again to the general theory, let R be any ring with identity and
, consider the set Zl of integral multiples of the dentity; stated symbolically,
Zl = {nlln E Z}. From the relations
nI - mI = (n - m)l, (nl)(ml) = (nm)l one can easily infer that ZI itself [orms a (commutative) ring with identity. The order of the additive cyclic group (Z1, +) is simply the characteristic oC the given ring R.
When R happens to be an integral domain, then Zl is a subdomain of R (that s, Z1 is also an integral domain with respect to the operations in R). In fact, ZI is the smallest subdomain oC R, in the sense that it is con-tained in every other subdomain of R. If R is a domain of characteristic p,
, I
PROBLEMS 13
where p is a prime, then we are able to deduce considerably more: each nonzero element of Zl is invertible. Before establishing this, first observe that the set ZI, regarded as an additive cyclic group of order p, consists of p distinct elements, namely, the p ~ums nI, where n = O, 1, ... , p 1. Now letnl beany nonzeroelementoCZ1 (O < n < p). Since nandp are relatively 'prime, there exist integers r and s Cor which rp + sn = 1. But then
1 = (rp + sn)1 = r(p1) + (s1)(nl). . As p1 = O, we obtain the equation 1 = (sl)(nl), so that sI serves as the multiplicative in verse of nI in ZL The value of these remarks wiIl have too
wait further developments (in particular, see Chapter 4).
;',.:PROBLEMS
::.' ). Verify that the zero elernent of a ring R is unique, as s the additive inverse of each.: element a E R. . '
. , .
). Let R be an additive commutative group: If the product of eveJ'Y pair of elernents is defined to be zero, show tbat tbe resulting systern f~rmiifa'coxirmutave ring " (this is sometimes called tbe zero ring). .
3. Prove that any ring R in which the two operations are equal (that is, a + b = ab for all a, b e R) must be the trivial ring R = {O}. .
4., In a ring R with identity, establish each of the following: a) !he identity elemen! for rnultiplieation is unique, b) if a e R has a rnultiplieative inverse, thena -1 is unique, e) ifthe'elernent a is invertible, then so. also is -a, d) no divisor of zero can Po.ssess a multiplica ti ve IDverse in R.
5. 'a) Ifthe set X eontains more than one elernent, prove that every nonempty proper subset of X is a zer divisor in the ring P(X).
b) Show that, if n > 1, the matrix ring Mn(R) has zero divison even though the ring R may not. ' _
6. Suppose !hat R is a ring with identity 1 'itnd having no divisor s ofzero. For a, bE R, verify that a) ab 1 if and only if ha = 1, b) if a2 1, then either a 1 or a = -1. 7~ Let a,' b be two elements ofthe ring R. Ji nE Z.: and a and b eommute, derive the
binomial expansion
(a + bY' = an + ('i.)an-1b + '" + (k)n-kbk + '" (n~l)Qb-l + b", where
[nj nI k "" k!(n - k)! is the usual binomial coefficent
..
12 FIRST COURSE IN RINGS AND IDEALS
a + O ofthe group (R, +) to have order m mean s that ma = O and ka + O irO < k < m.) But the retation O == ma = (ml)a mplies that mI = O, Cor R is assumed to be free of zero divisors. We therefore conclude from the theorem that n ::;; m, whence m and n are equal. In consequence, every nonzero element of R has additive order n.
A somewhat similar argumen't can be employed when char R O. The equaqon ma = O would lead, as before, to m1 = O or m O. In this case every nonzero eIement a E R mus be oC infinite order.
The last result seives to bring OUt anotner use fuI point, which we place on record as . .
CoroHary 2. only iC na
An integral dornain R has positive characteristic ifand O foro s?me O + a E R and some integer n E Z+.
Continuing in thls veln, let us next show tbat not any commutative group can serve as the additive group of an integral domain.
Theorem 1-6. The cbaracteristic of an integral domain is either zero or a prime number. '
Prooj. Let R be of positive characteristic n and assume that n is not a prime. Then, n has a nonrivial factorization n = nI n2 , with I < nI' n2 < n. It follows tbat
O = nl = (n 1n2)1 = (n In2)12 = (n I l)(n21). By supposition,.R is without zero divisors, so that either nll O or n2 I O. Since both nI and n2 are less than n, this contradicts the choice of n as the . leastpositive integer for which n1 = O. We therefore concluoe that char R must be prime.
CorolIary .. If R is a finite integral domain, then char R = .p, a prime. Turning again to the general theory, let R be any ring with identity and
, consider the set Zl of integral multiples of the dentity; stated symbolically,
Zl = {nlln E Z}. From the relations
nI - mI = (n - m)l, (nl)(ml) = (nm)l one can easily infer that ZI itself [orms a (commutative) ring with identity. The order of the additive cyclic group (Z1, +) is simply the characteristic oC the given ring R.
When R happens to be an integral domain, then Zl is a subdomain of R (that s, Z1 is also an integral domain with respect to the operations in R). In fact, ZI is the smallest subdomain oC R, in the sense that it is con-tained in every other subdomain of R. If R is a domain of characteristic p,
, I
PROBLEMS 13
where p is a prime, then we are able to deduce considerably more: each nonzero element of Zl is invertible. Before establishing this, first observe that the set ZI, regarded as an additive cyclic group of order p, consists of p distinct elements, namely, the p ~ums nI, where n = O, 1, ... , p 1. Now letnl beany nonzeroelementoCZ1 (O < n < p). Since nandp are relatively 'prime, there exist integers r and s Cor which rp + sn = 1. But then
1 = (rp + sn)1 = r(p1) + (s1)(nl). . As p1 = O, we obtain the equation 1 = (sl)(nl), so that sI serves as the multiplicative in verse of nI in ZL The value of these remarks wiIl have too
wait further developments (in particular, see Chapter 4).
;',.:PROBLEMS
::.' ). Verify that the zero elernent of a ring R is unique, as s the additive inverse of each.: element a E R. . '
. , .
). Let R be an additive commutative group: If the product of eveJ'Y pair of elernents is defined to be zero, show tbat tbe resulting systern f~rmiifa'coxirmutave ring " (this is sometimes called tbe zero ring). .
3. Prove that any ring R in which the two operations are equal (that is, a + b = ab for all a, b e R) must be the trivial ring R = {O}. .
4., In a ring R with identity, establish each of the following: a) !he identity elemen! for rnultiplieation is unique, b) if a e R has a rnultiplieative inverse, thena -1 is unique, e) ifthe'elernent a is invertible, then so. also is -a, d) no divisor of zero can Po.ssess a multiplica ti ve IDverse in R.
5. 'a) Ifthe set X eontains more than one elernent, prove that every nonempty proper subset of X is a zer divisor in the ring P(X).
b) Show that, if n > 1, the matrix ring Mn(R) has zero divison even though the ring R may not. ' _
6. Suppose !hat R is a ring with identity 1 'itnd having no divisor s ofzero. For a, bE R, verify that a) ab 1 if and only if ha = 1, b) if a2 1, then either a 1 or a = -1. 7~ Let a,' b be two elements ofthe ring R. Ji nE Z.: and a and b eommute, derive the
binomial expansion
(a + bY' = an + ('i.)an-1b + '" + (k)n-kbk + '" (n~l)Qb-l + b", where
[nj nI k "" k!(n - k)! is the usual binomial coefficent
..
14 FIRST COURSE IN RINGS AND IDEALS
8. An e1ement a of a ring R is said to be idempotent if a2 = a and nilpotent if a" = O for sorne n E Z+. Show that a) a nonzero idempotent element cannot be nilpotent, b) every nonzero nilpotent element is a zero divisor in R.
9. Given that R is an integral domain, prove that a) the only nilpotent element is the zero element of R, b) the multiplicative identity is the only nonzero idempotent elemento
10. If a is a nilpotent element of R, a ring with identity, establish that 1 + a is invertible in R. [Hint: (1 + a)-l = 1 - a + a2 + ... + (-l)"-la"-l, where a" = O.]
11. A Boo/ean ring is a ring with identity every element of which is idempotent. Prov~ that any Boolean ring R is cornmutative. [Hin!: First show that a = -a for every aE R.]
12. Suppose the ring R contains an element a such that (1) a is idempotent and (2) a is not a zero divisor of R. Deduce that a serves as a multiplicative identity for R.
13. Let S be a nonempty subset of the finite ring R. Prove that S is a subring of R if and only if S is c10sed under both the operations of addition and multiplication.
14. Assume that R is a ring l/lld a E R. If C(a) denotes the set of all elements which commute with a,
C(a) = {r E Rlar = ra}, show that C(a) js a ~ubring of R. Also, verify the equality tent R = n.on C(a).
1S. Given a ring R, prove that a) if SI is an arbitrary (indexed) coHection of subrings of R, then their intersection
n S is also a subring of R; b) for a nonempty subset T of R, the set
(T) = n {SIT S; S; S is a subring of R} .is the smaHest (in the sense of inc1usion) subring of R to contain T; (T) is called the subring generated by T.
16. Let S be a subring of R, a ring with identity. For an arbitrary element a rt S, the subring generated by the set S u {a} is represented by (S, a). If a E cent R, establish tha t
(S, a) = {ro + rla + ... + r.a"ln E Z+; r E S}. 17. Let R be an arbitrary ring apd n E Z+. Ifthe set Sft is defined by
S. = {a E Rlnka = O for sorne k> O}, determine whether S" is a subring of R ..
18. Establish the foHowing assertions concerning the characteristic of a ring R:
PROBLEMS 15
a) if there exists an integer k such that ka = O for aH a E R, then k is divisible by char R;
b) if char R :> O, then char S ~ char R for any subring S of R; c) if R is an integral domain and. S is a subdomain of R, then char S = char R.
19. L:t R be a ring with a finite number of elements, sayal> a2' ... , ano and let ni be the order of ai regarded as a member of the additive group of R. Prove that the characteristic of R is the least common multiple of the integers ni (i = 1,2, ... , n).
20. Suppose that R is a ring with identity such that char R = n > O. If n is not prime, show that R has divisors of zero.
21. If R is a rillg which has no nonzero nilpotent elements, deduce that aH the idem-potent elements of R belong to cent R. [Hint: If a2 = a, then (ara - arV = (ara - ra)2 = O for aH r E R.]
22. Assume that R is a ring with the property that a2 + a E cent R for every element a in R. Prove that R is necessarily a cornmutative ringo [Hint: Utilize the expression (a + W + (a + b) to show first that ab + ba lies in the center for aH a, b E R.]
23. Let (G, +) be a commutative group and R be the set ofall (group) homomorphisms of G into itself. The pointwise sum f + g and composition f o g of two functions f, g E R are defined by the usual rules
(f + g)(x) = f(x) + g(x), (f o g)(x) = f(g(x) (XE G). Show that the resulting system (R, +, o) forms a ring. At the same time determine the invertible elements of R.
24. Let(G,') bea finite group (writtenmultiplicatively), say with elementsx, x 2 , , x"' and let R be an arbitrary ringo Consider the set R(G) of all formal sums
" rxi (rER). i=l
Two such expressions are regarded as equal if they have the same coefficients .. Addition and multiplication can be defined in R(G) by taking
rx + sx = (r + s)x i=l i=l i=l
and (. rx) [. SX) = . tx j , l=1 ,=1 1=1 where
ti = rjsk XJXk=Xf
(The meaning of the last-written sum is that the surnmation is to be extended over aH subscripts j and k for which xjxk = x.) Prove that, with respect to these operations, R(G) constitutes a ring, the so-called group ring of G over R.
14 FIRST COURSE IN RINGS AND IDEALS
8. An e1ement a of a ring R is said to be idempotent if a2 = a and nilpotent if a" = O for sorne n E Z+. Show that a) a nonzero idempotent element cannot be nilpotent, b) every nonzero nilpotent element is a zero divisor in R.
9. Given that R is an integral domain, prove that a) the only nilpotent element is the zero element of R, b) the multiplicative identity is the only nonzero idempotent elemento
10. If a is a nilpotent element of R, a ring with identity, establish that 1 + a is invertible in R. [Hint: (1 + a)-l = 1 - a + a2 + ... + (-l)"-la"-l, where a" = O.]
11. A Boo/ean ring is a ring with identity every element of which is idempotent. Prov~ that any Boolean ring R is cornmutative. [Hin!: First show that a = -a for every aE R.]
12. Suppose the ring R contains an element a such that (1) a is idempotent and (2) a is not a zero divisor of R. Deduce that a serves as a multiplicative identity for R.
13. Let S be a nonempty subset of the finite ring R. Prove that S is a subring of R if and only if S is c10sed under both the operations of addition and multiplication.
14. Assume that R is a ring l/lld a E R. If C(a) denotes the set of all elements which commute with a,
C(a) = {r E Rlar = ra}, show that C(a) js a ~ubring of R. Also, verify the equality tent R = n.on C(a).
1S. Given a ring R, prove that a) if SI is an arbitrary (indexed) coHection of subrings of R, then their intersection
n S is also a subring of R; b) for a nonempty subset T of R, the set
(T) = n {SIT S; S; S is a subring of R} .is the smaHest (in the sense of inc1usion) subring of R to contain T; (T) is called the subring generated by T.
16. Let S be a subring of R, a ring with identity. For an arbitrary element a rt S, the subring generated by the set S u {a} is represented by (S, a). If a E cent R, establish tha t
(S, a) = {ro + rla + ... + r.a"ln E Z+; r E S}. 17. Let R be an arbitrary ring apd n E Z+. Ifthe set Sft is defined by
S. = {a E Rlnka = O for sorne k> O}, determine whether S" is a subring of R ..
18. Establish the foHowing assertions concerning the characteristic of a ring R:
PROBLEMS 15
a) if there exists an integer k such that ka = O for aH a E R, then k is divisible by char R;
b) if char R :> O, then char S ~ char R for any subring S of R; c) if R is an integral domain and. S is a subdomain of R, then char S = char R.
19. L:t R be a ring with a finite number of elements, sayal> a2' ... , ano and let ni be the order of ai regarded as a member of the additive group of R. Prove that the characteristic of R is the least common multiple of the integers ni (i = 1,2, ... , n).
20. Suppose that R is a ring with identity such that char R = n > O. If n is not prime, show that R has divisors of zero.
21. If R is a rillg which has no nonzero nilpotent elements, deduce that aH the idem-potent elements of R belong to cent R. [Hint: If a2 = a, then (ara - arV = (ara - ra)2 = O for aH r E R.]
22. Assume that R is a ring with the property that a2 + a E cent R for every element a in R. Prove that R is necessarily a cornmutative ringo [Hint: Utilize the expression (a + W + (a + b) to show first that ab + ba lies in the center for aH a, b E R.]
23. Let (G, +) be a commutative group and R be the set ofall (group) homomorphisms of G into itself. The pointwise sum f + g and composition f o g of two functions f, g E R are defined by the usual rules
(f + g)(x) = f(x) + g(x), (f o g)(x) = f(g(x) (XE G). Show that the resulting system (R, +, o) forms a ring. At the same time determine the invertible elements of R.
24. Let(G,') bea finite group (writtenmultiplicatively), say with elementsx, x 2 , , x"' and let R be an arbitrary ringo Consider the set R(G) of all formal sums
" rxi (rER). i=l
Two such expressions are regarded as equal if they have the same coefficients .. Addition and multiplication can be defined in R(G) by taking
rx + sx = (r + s)x i=l i=l i=l
and (. rx) [. SX) = . tx j , l=1 ,=1 1=1 where
ti = rjsk XJXk=Xf
(The meaning of the last-written sum is that the surnmation is to be extended over aH subscripts j and k for which xjxk = x.) Prove that, with respect to these operations, R(G) constitutes a ring, the so-called group ring of G over R.
TWO
IDEALS AND THEIR OPERATIONS
A1though it is possibleto obtain sorne interesting conclusions conceming subrings, this concept,lifunrestricted, is too gen
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