Elements of mathematics: algebra I, Òîì 2

NICOLAS BOURBAKI

Elements of Mathematics

Algebra I Chapters 1-3

* HERMANN, PUBLISHERS IN ARTS AND SCIENCE

293 rue Lecourbe, 75015 Paris, France A ▼▼ ADDISON-WESLEY PUBLISHING COMPANY

Advanced Book Program Reading, Massachusetts

Originally published as ELEMENTS DE MATHEMATIQUE, ALGEBRE © 1943, 1947, 1948, 1971 by Hermann, Paris

ISBN 2-7056-5675-8 (Hermann) ISBN 0-201-00639-1 (Addison-Wesley) Library of Congress catalog card number LC 72-5558 American Mathematical Society (MOS) Subject Classification Scheme (1970): I5-A03, 15-A69, 15-A75, 15-A78 Printed in Great Britain

© 1974 by Hermann. All rights reserved This book, or parts thereof, may not be reproduced in any form without the publisher’s written permission

TO THE READER

1. This series of volumes, a list of which is given on pages ix and x, takes u p mathematics at the beginning, and gives complete proofs. In principle, it requires no particular knowledge of mathematics on the readers’ part, but only a certain familiarity with mathematical reasoning and a certain capacity for abstract thought. Nevertheless, it is directed especially to those who have a good knowledge of at least the content of the first year or two of a university mathematics course. 2. The method of exposition we have chosen is axiomatic and abstract, and normally proceeds from the general to the particular. This choice has been dictated by the main purpose of the treatise, which is to provide a solid foundation for the whole body of modem mathematics. For this it is indis­ pensable to become familiar with a rather large number of very general ideas and principles. Moreover, the demands of proof impose a rigorously fixed order on the subject matter. It follows that the utility of certain considerations will not be immediately apparent to the reader unless he has already a fairly extended knowledge of mathematics; otherwise he must have the patience to suspendjudgment until the occasion arises. 3. In order to mitigate this disadvantage we have frequently inserted examples in the text which refer to facts the reader may already know but which have not yet been discussed in the series. Such examples are always placed between t w o asterisks: • .. . Most readers will undoubtedly find that these examples will help them to understand the text, and will prefer not to leave them out, even at a first reading. Their omission would of course have no disadvantage, f r o m a purely logical point of view. 4. This series is divided into volumes (here called "Books”). The first six B o o k s are numbered and, in general, every statement in the text assumes as known only those results which have already been discussed in the preceding v

TOTHEREADER

volumes. This rule holds good within each Book, but for convenience of expo­ sition these Books are no longer arranged in a consecutive order. At the begin­ ning of each of these Books (or of these chapters), the reader will find a precise indication of its logical relationship to the other Books and he will thus be able to satisfy himself of the absence of any vicious circle. 5. The logical framework of each chapter consists of the definitions, the axioms, and the theorems of the chapter. These are the parts that have mainly to be borne in mind for subsequent use. Less important results and those which can easily be deduced from the theorems are labelled as "propositions”, "lemmas”, "corollaries”, "remarks”, etc. Those which may be omitted at a first reading are printed in small type. A commentary on a particularly important theorem appears occasionally under the name of “scholium”. T o avoid tedious repetitions it is sometimes convenient to introduce nota­ tions or abbreviations which are in force only within a certain chapter or a certain section of a chapter (for example, in a chapter which is concerned only with commutative rings, the word "ring” would always signify "commutative ring”). Such conventions are always explicitly mentioned, generally at the beginning of the chapter in which they occur. 6. Some passages in the text are designed to forewarn the reader against serious errors. These passages are signposted in the margin with the sign

^2 . ("dangerous bend”). 7. The Exercises are designed both to enable the reader to satisfy himself that he has digested the text and to bring to his notice results which have no place in the text but which are nonetheless of interest. The most difficult exercises bear the sign ^J. 8 . In general, we have adhered to the commonly accepted terminology, except where there appeared to be good reasons for deviating from it.

9. We have made a particular effort always to use rigorously correct language, without sacrificing simplicity. As far as possible we have drawn attention in the text to abuses <S language, without which any mathematical text runs the risk of pedantry, not to say unreadability. 10. Since in principle the text consists of the dogmatic exposition of a theory, it contains in general no references to the literature. Bibliographical references are gathered together in Historical Notes, usually at the end of each chapter. These notes also contain indications, where appropriate, of the unsolved problems of the theory. The bibliography which follows each historical note contains in general only those books and original memoirs which have been of the greatest impor­ tance in the evolution of the theory under discussion. It makes no sort of pre­

I

TO THE READER

tence to completeness; in particular, references which serve only to determine questions of priority are almost always omitted. As to the exercises, we have not thought it worthwhile in general to indicate their origins, since they have been taken from many different sources (original papers, textbooks, collections of exercises). 11. References to a part of this series are given as follows: a) If reference is made to theorems, axioms, or definitions presented in the same section, they are quoted by their number. b) If they occur in another section of the same chapter, this section is also quoted in the reference. c) If they occur in another chapter in the same Book, the chapter and section are quoted. d) If they occur in another Book, this Book is first quoted by its title. The Summaries of Results are quoted by the letter R; thus Set Theory, R signifies “Summary of Results cf the Theory of Sets”.

CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES

Lie groups and lie algebras

1 . Lie algebras. 2. Free Lie algebras. 3. Lie groups. 4. Coxeter groups and Tits systems. 5. Groups generated by reflections. 6 . Root systems. 0QV1MUIXI1VE algebra

1. Flat modules. 2. Localization. 3. Graduations, filtrations. logies. 4. Associated prime ideals and primary decomposition. 5. Integers. 6 . Valuations. 7. Divisors.

S PEC TRALTHEORY 1. Normed algebras. 2. Locally compact groups. Differential and Analytic Manifolds

Summary of results.

x

and

topo­

CONTENTS

To the Reader................................................................................................................

v

Contents of the Elements of Mathematics Series.........................................................

jx

Introduction............................................................................................................. Chapter I. Algebraic Structures..........................................................................................

§ 1. Laws of composition; associativity; commutativity.................................. 1. Laws of composition ................................................................. 2. Composition of an ordered sequence of elements............................ 3. Associative laws......................................................................... 4. Stable subsets. Induced laws...................................................... 5. Permutable elements. Commutative laws........................................ 6 . Quotient laws ............................................................................ $2. Identity element; cancellable elements; invertible elements .. 1. Identity element ........................................................................ 2 . Cancellable elements ................................................................ 3. Invertible elements..................................................................... 4. Monoid of fractions of a commutative monoid................................ 5. Applications : I. Rational integers.............................................. 6. Applications: II. Multiplication of rational integers......................... 7. Applications: III. Generalized powers ............................................ 8 . Notation .................................................................................... § 3. Actions................................................................................................ 1. Actions ...................................................................................... 2. Subsets stable under an action. Induced action................................. 3. Quotient action ......................................................................... 4. Distributivity.............................................................................. 5. Distributivity of one internal law with respect to another.

xxi

\ \

1 3 4 6

7 11

12 12 14 15 17

20 22 23 23 24 24 9g 9g 97 99

CONTENTS

§ 4. Groups and groups with operators ..................................................... 30 1. Groups ...................................................................................... 30 2- Groups with operators ................................................................ 31 3. Subgroups ................................................................................. 32 4. Quotient groups ........................................................................ 34 5. Decomposition of a homomorphism ................................................. 37 6 . Subgroups of a quotient group ................................................. 38 7- The Jordan-Holder theorem ....................................................... 41 8 . Products and fibre products....................................................... 45 9. Restricted sums......................................................................... 47 48 10. Monogenous groups .................................................................. § 5. Groups operating on a set.................................................................... 1. Monoid operating on a set.......................................................... 2. Stabilizer, fixer.......................................................................... 3. Inner automorphisms................................................................. 4. Orbits......................................................................................... 5. Homogeneous sets .................................................................... 6 . Homogeneous principal sets...................................................... 7. Permutation groups of a finite set ............................................ § 6 . Extensions, solvable groups, nilpotent groups........................................... 65 1. Extensions ................................................................................ 2. Commutators............................................................................. 3. Lower central series, nilpotent groups............................................... 4. Derived series, solvable groups ................................................ 5. />-groups ................................................................................... 6 . Sylow subgroups....................................................................... 7. Finite nilpotent groups...............................................................

52 52 54 55 56 58 60 61

65 68

71 74 76 78 80

§ 7. Free monoids, free groups ................................................................. 81 1. Free magmas............................................................................. 81 2. Free monoids ............................................................................ 82 3. Amalgamated sum of monoids ................................................. 84 4. Application to free monoids ..................................................... 88 5. Free groups ............................................................................... 89 6 . Presentations of a group ........................................................... 90 7. Free commutative groups and monoids ............................................ 92 8 . Exponential notation ................................................................. 94 9. Relations between the various free objects ...................................... 95 $ 8 . Rings ................................................................................................. 1. Rings.......................................................................................... 2 ■ Consequences of distributivity.................................................. xii

96 96 98

CONTENTS

3. Examples of rings ..................................................................... ........... 1^1 4. Ring homomorphisms............................................................................102 5. Subrings.................................................................................................103 6 . Ideals ........................................................................................ ............103 7. Quotient rings............................................................................ ............105 8 . Subrings and ideals in a quotient ring............................................... ... 106 9. Multiplication of ideals ........................................................................107 10. Product of rings ........................................................................ ...........108 11. Direct decomposition of a ring ................................................. ...........HO 12. Rings of fractions ..................................................................... ...........112 §9. Fields.................................................................................................... .......... 114 1. Fields ................................................................................................... 114 2. Integral domains................................................................................... 116 3. Prime ideals ......................................................................................... 116 4. The field of rational numbers .................................................... .......... 117 § 10. Inverse and direct limits...................................................................... 1. Inverse systems of magmas ...................................................... ..........118 2. Inverse limits of actions ......................................................................H9 3. Direct systems of magmas .................................................................. 120 4. Direct limit of actions ......................................................................... 123 Exercises for § 1 ........................................................................................ ......... 124 Exercises for $ 2 ................................................................................................. 126 Exercises for $ 3 ........................................................................................ 129 Exercises for § 4 ........................................................................................ 132 Exercises for $ 5 ........................................................................................ 140 Exercises for § 6 ........................................................................................ 147 Exercises for § 7 ........................................................................................ 159 Exercises for § 8 ........................................................................................ 171 Exercises for §9 ......................................................................................... 174 Exercises for § 10 ..................................................................................... 179 Historical note ........................................................................................... 180 Chapter II. Linear Algebra ......................................................................... 191 $ 1. Modules.............................................................................................. 191 1. Modules; vector spaces; linear combinations................................... 191 2. Linear mappings ....................................................................... 194 3. Submodules; quotient modules ................................................. 196 xm

CONTENTS

4. Exact sequences ........................................................................ 5. Products of modules................................................................... 6 . Direct sum of modules............................................................... 7. Intersection and sum of submodules.................................................. 8 . Direct sums of submodules........................................................ 9. Supplementary submodules ...................................................... 10. Modules of finite length ............................................................ 11. Free families. Bases .................................................................. 12. Annihilators. Faithful modules. Monogenous modules ... 13. Change of the ring of scalars..................................................... 14. Multimodules............................................................................. § 2. Modules of linear mappings. Duality ................................................. 1. Properties of HomA(E, F) relative to exact sequences .... 2. Projective modules .................................................................... 3. Linear forms; dual of a module ................................................. 4. Orthogonality ............................................................................ 5. Transpose of a linear mapping .................................................. 6 . Dual of a quotient module. Dual of a direct sum. Dual bases 7. Bidual ........................................................................................ 8 . Linear equations......................................................................... § 3. Tensor products .................................................................................. 1. Tensor product of two modules................................................. 2. Tensor product of two linear mappings............................................. 3. Change of ring........................................................................... 4. Operators on a tensor product; tensor products as multi­ modules ................................................................................. 5. Tensor product of two modules over a commutative ring 6 . Properties of E ®A F relative to exact sequences.............................. 7. Tensor products of products and direct sums................................... 8 . Associativity of the tensor product ............................................ 9. Tensor product of families of multimodules .....................................

200 202 205 208 210 212 214 219 221 224 227 227 231 232 234 234 236 239 240 243 243 245 246 247 249 251 254 258 259

§ 4. Relations between tensor products and homomorphism modules 267 1. The isomorphisms HomB(E(g>A F. G)-* HomA(F, HomB(E, G)) and Hom0(E HomA(E, F)............................... 268 3. Trace of an endomorphism........................................................ 273 4. The homomorphism Homc(E1, Fj) A U B between subsets of a set E; a law of composition (21, 93) t-> F(9l, 93) is derived between subsets of ^(E), F(2I, S3) being the set of A l l B with A e 21, B e 23; but F(2t, 23) should not be denoted by 21 KJ 23, as this notation already has a different meaning ( the union of 21 and 23 considered as subsets of ^P(E)).

2

COMPOSITION OF A N ORDERED SEQUENCE OF ELEMENTS

For a mapping f of E into E’ to be an isomorphism, it is necessary and -l sufficientthat it be a bijective homomorphism and f is then an isomorphism of E' onto E. 2. COMPOSITION OF AN ORDERED SEQUENCE OF ELEMENTS

Recall that afamily of elements of a set E is a mapping t of a set I (called an indexing set) into E; a family (*t)lsI is called finite if the indexing set isfinite. A finite family (*JleI of elements of E whose indexing set I is totally ordered is called an ordered sequence of elements of E. In particular, every finite sequence (*i)ieH> where H is a finite subset of the set N of natural numbers, can be considered as an ordered sequence if H is given the order relation induced by the relation m < n between natural numbers. Two ordered sequences (*t)ieI and ( y k ) k e K are called similar if there exists an ordered set isomorphism <j) of I onto K such that ym> = xt for all i e I. Every ordered sequence (xa)aeA is similar to a suitable finite sequence. For there exists an increasing bijection of A onto an interval [0, n] of N. Definition 4. Let {xa)aeA be an ordered sequence of elements in a magma E whose

indexing set A is non-empty. The composition (under the law T) of the ordered sequence (*a)aeA> denoted by T xa, is the element of E defined by induction on the number of ' aeA

'

elements in A as fallows:

(1 ) if A = {(3} then T xa = xa; a e A

(2) if A has p > 1 elements, (3 is the least element of A and A’ = A {(3), then

T Xa Xq t(T

n c A

\a e A / •

It follows immediately (by induction on the number of elements in the in­ dexing sets) that the compositions of two similar ordered sequences are equal; in particular, the composition of any ordered sequence is equal to the composi­ tion of a finite sequence. If A = [A, y., v} has three elements (A < [x < v) the composition T xa is xK T T *v). a e A

Remark. Note that there is a certain arbitrariness about the definition of the composition of an ordered sequence; the induction we introduced proceeds “from right to left”. If we proceeded "from left to right”, the composition of the above ordered sequence (**., x^, ?cv) would be (x>t T xv) T xy.

As a matter of notation, the composition of an ordered sequence (xa)aeA is written a-j-A for a law denoted by 1 ; for a law written additively it is usually denoted by xa and called the sum of the ordered sequence (xa)aeA (the xa being called the terms of the sum); for a law written multiplicatively it is usually

ALGEBRAIC STRUCTURES

I

denoted by nA*. and called the product of the ordered sequence (*a) (the xa being called thefactors of the product).? When there is no possible confusion over the indexing set (nor over its ordering) it is often dispensed with in the notation for the composition of an ordered sequence and we then write, for example for a law written additively, 2 xa instead of 2 xa\ similarly for the other notations, aeA

For a law denoted by T the composition of a sequence (xf) with indexing set a 4

non-empty interval (fi- of N is denoted by T, ^ xi or ^ xt; similarlyfor laws denoted by other symbols. Let E and F be two magmas whose laws of composition are denoted by T andf a homomorphism of E into F. For every ordered sequence (xa,)aeA of elements of E

(2)

/(\a T ) = T/(*«). e A/ a e A

x7

3. ASSOCIATIVE LAWS Definition 5. A law of composition for all elements x, y, z in E,

(x, y) h>

x

T y on a set E is called associative if,

(xTy) T z = x T {y T z). A magma whose law is associative is called an associative magma.

The opposite law of an associative law is associative. Examples. (1) Addition and multiplication of natural numbers are associative laws ofcomposition onN (Set Theory, III, § 3, no. 3, Corollary to Proposition5)

(2) The laws cited in Examples (1), (3) and (4)of no. 1 are associative. Theorem 1 (Associativity theorem). Let E be an associative magma whose law is deflated by T. Let A be a totally ordered non-empty fin^e set, which is the union of an

ordered sequence of non-empty subsets (Bj)jsI such that the relations a e B(, P 6 Bj5 i < j imply a < (3; let (xa)a e A /*> an ordered sequence cf elements in E with A as index­ ing set. Then

(3) w

T *a = T<el(T *„) VxeB /

aeA

t

t The use of this terminology and the notation n Xx must be avoided if there is any risk of confusion with the product of the sets xa defined in the theory of sets (Set Theoy, II, § 5, no. 3). However, if the xa are cardinals and addition (resp. multipli­ cation) is the cardinal sum (resp. the cardinal product), the cardinal denoted by 2 xa (resp. FI xa) in the above notation is the cardinal sum (resp. cardinal pro­ duct) of the family (*.)«* (Set Theory, HI, 8 3, no. 3).

ASSOCIATIVE LAWS

§ 1.3

We prove the theorem by induction on the cardinal n of A. Let p be the cardinal ofl and/jits least element; letJ = 1 — jhj. If n = 1, as the B, are non­ empty, of necessityp = 1 and the theorem is obvious. Otherwise, assuming the theorem holds for an indexing set with at most n — 1 elements, we distinguish two cases: (a) Bh has a single element (3. Let C = U B(. The left-hand side of (3) is equal, by definition, to xB T xm\; the right-hand side is equal, by defini­ tion, to

* 3 T IleJ (T\ae(Bj T * a )); // the result follows from the fact that the theorem is assumed true for C and (B() fej(b) Otherwise, let (3 be the least element of A (and hence of Bh); let A’ = A — {“} and let B(' = A’ n B, for t e l ; then BJ = B, for i e J. The set A‘ has n — 1 elements and the conditions of the theorem hold for A and its sub­ sets B,'; hence by hypothesis:

Forming the composition of xa with each side, we have on the left-hand side side, by definition, T xa and on the right-hand side, using the associativity, aeA

which is equal, by Definition 3, to the right-hand side of formula (3). For an associative law denoted by T the composition y ^ of a sequence (A:i)(s|p,

<j| is also denoted (since no confusion can arise) bPy 1 f q

A particular case of Theorem 1 is the formula *o T *i T • • • T xn = (Xq T * x T . . • T *n-i) T xn. Consider an ordered sequence of n terms all of whose terms are equal to the same element # e E. The composition of this sequence is denoted by T * for a

n

law denoted by T, -L x for a law denoted by j_. For a law written multiplica­ tively the compositionis denoted by xn and called the n-th power of x . For a law written additively the composition is usually denoted by nx. The associativity 5

j

ALGEBRAIC STRUCTURES

theorem applied to an ordered sequence all of whose terms are equal gives the equation n i + 112 + • . . + n p / « i \ / n 2 \

(tip \

T x = VT x) T It x) T • ■ ■ T \T x).

In particular, if p =2, m + n

(4)

/ m \ / n \

T x = \T x) T \T x)

and if n± = n2 = ... = nf = m, pm

(5)

p / m \

T * = T \r x).

If X is a subset of E, we sometimes denote, in conformity with the above nota­ tion, by T X the set Xj T X2 T . ■- T Xp, where Xi = X2 = . . . = Xp - X ; it is thus the set of all compositions*! T x2 T • • • T xp with xx e X, x2 e X, ...

3

xp e X.

p

It is important not to confuse this set with the set of T x, where x runs through X. 4 STABLE SUBSETS. INDUCED LAWS Definition 6. A subset A of a set E is called stable under the law of composition t on E

if the composition of two elements of A belongs to A. The mapping (x,y ) x T y of A x A into A is then called the law induced on A by the law T. The set A with the law induced by T is called a submagma of E.

I n other words, for A to be stable under a law T it is necessary and sufficient that A T A c A. A stable subset of E and the corresponding submagma are often identified. The intersection of a family of stable subsets of E is stable; in particular there exists a smallest stable subset A of E containing a given subset X; it is said to be generated by X and X is called a generating system of A or a generating set of A. The corresponding submagma is also said to be generated by X. Proposition 1. Let E afid F be two magmas andf a homomorphism of E into F. (i) The image underf of a stable subset 1 and suppose the theorem is true when Card A < p. We prove it for Card A = p. It may be assumed that A is the interval (0, p — 1) in N; the composition of the ordered sequence (*«)«£ a defined by the natural order relation on A is T xi­

i= 0 Let A be given another total ordering and let h be the least element of A under this ordering and A’ the set of other elements of A (totally ordered by the induced ordering). Suppose first 0 < h < p — 1 and let P ={0, 1, . . . , h — 1 } and Q = ( h + 1, ... ,p — 1}; the theorem being assumed true forA', applying the associativity theorem, we obtain (since A' = Pu Q_)

T * = ( TV) t ( T1 *«) as A' a \l = 0 / Vi = h +1 / whence, composing xh with both sides and repeatedly applying the commuta­ tivity and associativity of T : “£A

T*a = Xh T ( T xa) = xn T ( f *,) T ( f x) \a e A ' /

/h-1 \

\ i = 0 / \i = h + l /

/ P - l

\

p - 1

= I T *,) T ** T ( T AT,) = T x„

\i = o /

\( = h + i 7

(

= o

1

which proves the theorem for this case. If h = 0 or h = p — 1, the same result follows, but more simply, the terms arising from P or the terms arising from Q not appearing in the formulae. Under a commutative associative law on a set E the composition of a finite family (*a)aeA of elements of E is by definition the common value of the com­ position of all the ordered sequences obtained by totally ordering A in all possible ways. This composition will still be denoted by T xa under a law denoted by T J similarly for other notations.

“6A

Theorem 3. Let T be an associative law on E and (*a)a a a non-emptyfinitefamily of dements of E which are pairwise permutable. If A is ft union of non-empty subsets ft),* i which are pairwise disjoint, then (6)

“Ta*° = ,TiC7Bi4 9

I

ALGEBRAIC STRUCTURES

This follows from Theorem 2 if A and I are totally ordered so that the Bf satisfy the conditions of Theorem 1. We single out two important special cases of this theorem: 1 If (^qcb)

aeA

aeA'

where A is the totally ordered set derived f r o m A by replacing the order on A by the opposite order.

This corollary follows from Proposition 4 by induction on the number of elements in A. In particular, if x and x' are inverses, T? x and "F x are inverses for every integer n ^ 0 . Corollary 2.

In a monoid the set of invertible elements is stable.

Proposition 5. If in a monoid x and x' are inverses and x commutes w i t h y , so does x'.

From =y T x, we deduce x' T (x Ty) T x' = x’ T ( yT x) T x’ and hence (x’ T x) T ( y T x’) = (x’ T y ) T (x T x ’ ) , that is tT *’ = x’ T y . 1. Let E be a monoid, X a subset ofH and X’ the centralizer qf'X.. The inverse of every invertible element ofX‘ belongs to X’.

Corollary

Corollary 2. In a monoid the inverse