Basic-Algebra-II-2E-by-Nathan-Jacobson

Basic Algebra II
Second Edition
NATHAN

JACOBSON

YALE U N I V E R S I T Y

•B

W . H. F R E E M A N A N D

COMPANY

N e w York

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To Mike and Polly

Library of Congress Cataloging-in-Publication Data
(Revised for vol. 2)
Jacobson, Nathan, 1910Basic algebra.
Includes bibliographical references and indexes.
1. Algebra. I. Tide.
QA154.2.J32 1985
512.9
84-25836
ISBN 0-7167-1480-9 (v. 1)

ISBN 0-7167-1933-9 (v. 2)
Copyright © 1989 by W. H. Freeman and Company
No part of this book may be reproduced by any
mechanical, photographic, or electronic process, or in
the form of a phonographic recording, nor may it be
stored in a retrieval system, transmitted, or otherwise
copied for public or private use, without written
permission from the publisher.
Printed in the United States of America
1 2 3 4 5 6 7 8 9 0 VB 7 6 5 4 3 2 1 0 8 9

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Contents

Contents of Basic Algebra
Preface

I

x

xiii

Preface to the First Edition

INTRODUCTION

0.1

0.2
0.3
0.4

1

2

1

Zorn's lemma 2
Arithmetic of cardinal numbers 3
Ordinal and cardinal numbers 4
Sets and classes 6
References 7

CATEGORIES

1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8

8

Definition and examples of categories 9

Some basic categorical concepts 15
Functors and natural transformations 18
Equivalence of categories 26
Products and coproducts 32
The horn functors. Representable functors 37
Universals 40
Adjoints 45
References 51

UNIVERSAL ALGEBRA

2.1
2.2
2.3
2.4

xv

52

ft-algebras 53
Subalgebras and products 58
Homomorphisms and congruences 60
The lattice of congruences. Subdirect products

66

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vi

Conten

2.5
2.6
2.7
2.8
2.9
2.10

3

MODULES

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15

4

94

The categories R-mod and mod-R 95
Artinian and Noetherian modules 100
Schreier refinement theorem. Jordan-Holder theorem 104
The Krull-Schmidt theorem 110
Completely reducible modules 117
Abstract dependence relations. Invariance of dimensionality
Tensor products of modules 125
Bimodules 133
Algebras and coalgebras 137
Projective modules 148
Injective modules. Injective hull 156
Morita contexts 164
The Wedderburn-Artin theorem for simple rings 171
Generators and progenerators 173
Equivalence of categories of modules 177
References 183

B A S I C S T R U C T U R E T H E O R Y OF RINGS

4.1
4.2
4.3
4.4
4.5
4.6

4.7
4.8

5

Direct and inverse limits 70
Ultraproducts 75
Free Q-algebras 78
Varieties 81
Free products of groups 87
Internal characterization of varieties 91
References 93

184

Primitivity and semi-primitivity 185
The radical of a ring 192
Density theorems 197
Artinian rings 202
Structure theory of algebras 210
Finite dimensional central simple algebras 215
The Brauer group 226
Clifford algebras 228
References 245

CLASSICAL REPRESENTATION THEORY
OF FINITE G R O U P S

5.1
5.2

5.3
5.4

246

Representations and matrix representations of groups 247
Complete reducibility 251
Application of the representation theory of algebras 257
Irreducible representations of S 265
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vii

Contents

5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14

5.15

Characters. Orthogonality relations 269
Direct products of groups. Characters of abelian groups 279
Some arithmetical considerations 282
Burnside's p q theorem 284
Induced modules 286
Properties of induction. Frobenius reciprocity theorem 292
Further results on induced modules 299
Brauer's theorem on induced characters 305
Brauer's theorem on splitting fields 313
The Schur index 314
Frobenius groups 317
References 325
a

b

ELEMENTS OF H O M O L O G I C A L A L G E B R A
WITH APPLICATIONS

6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9

6.10
6.11
6.12
6.13

7

326

Additive and abelian categories 327
Complexes and homology 331
Long exact homology sequence 334
Homotopy 337
Resolutions 339
Derived functors 342
Ext 346
Tor 353
Cohomology of groups 355
Extensions of groups 363
Cohomology of algebras 370
Homological dimension 375
Koszul's complex and Hilbert's syzygy theorem
References 387

378

C O M M U T A T I V E IDEAL THEORY: GENERAL THEORY
A N D NOETHERIAN RINGS

7.1

7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11

388

Prime ideals. Nil radical 389
Localization of rings 393
Localization of modules 397
Localization at the complement of a prime ideal.
Local-global relations 400
Prime spectrum of a commutative ring 403
Integral dependence 408
Integrally closed domains 412
Rank of projective modules 414
Projective class group 419
Noetherian rings 420
Commutative artinian rings 425

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Contents

7.12
7.13
7.14
7.15
7.16
7.17
7.18

8

FIELD T H E O R Y

8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
8.11
8.12
8.13
8.14
8.15

8.16
8.17
8.18
8.19

9

Affine algebraic varieties. The Hilbert Nullstellensatz 427
Primary decompositions 433
Artin-Rees lemma. Krull intersection theorem 440
Hilbert's polynomial for a graded module 443
The characteristic polynomial of a noetherian local ring 448
Krull dimension 450
J-adic topologies and completions 455
References 462
463

Algebraic closure of a field 464
The Jacobson-Bourbaki correspondence 468
Finite Galois theory 471
Crossed products and the Brauer group 475
Cyclic algebras 484
Infinite Galois theory 486
Separability and normality 489
Separable splitting fields 495
Kummer extensions 498
Rings of Witt vectors 501
Abelian p-extension 509
Transcendency bases 514
Transcendency bases for domains. Affine algebras 517

Luroth's theorem 520
Separability for arbitrary extension fields 525
Derivations 530
Galois theory for purely inseparable extensions of exponent one 541
Tensor products of fields 544
Free composites of fields 550
References 556

VALUATION THEORY

9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
9.10
9.11
9.12

557

Absolute values 558
The approximation theorem 562
Absolute values on Q and F(x) 564
Completion of a field 566
Finite dimensional extensions of complete fields.

The archimedean case 569
Valuations 573
Valuation rings and places 577
Extension of homomorphisms and valuations 580
Determination of the absolute values of a finite
dimensional extension field 585
Ramification index and residue degree. Discrete valuations 588
Hensel's lemma 592
Local fields 595

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ix

Contents

9.13
9.14
9.15

10

DEDEKIND D O M A I N S

10.1
10.2
10.3
10.4
10.5

10.6

11

Totally disconnected locally compact division rings 599
The Brauer group of a local field 608
Quadratic forms over local fields 611
References 618
619

Fractional ideals. Dedekind domains 620
Characterizations of Dedekind domains 625
Integral extensions of Dedekind domains 631
Connections with valuation theory 634
Ramified primes and the discriminant 639
Finitely generated modules over a Dedekind domain
References 649

F O R M A L L Y REAL FIELDS

11.1
11.2
11.3
11.4
11.5
11.6
11.7

643

650

Formally real fields 651
Real closures 655
Totally positive elements 657
Hilbert's seventeenth problem 660
Pfister theory of quadratic forms 663
Sums of squares in R(x ..., x ), R a real closed field 669
Artin-Schreier characterization of real closed fields 674
References 677
u

INDEX

n

679

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Contents
of Basic A l g e b r a I

I N T R O D U C T I O N : C O N C E P T S F R O M SET T H E O R Y .
THE INTEGERS

0.1
0.2
0.3

0.4
0.5
0.6
0.7
1

The power set of a set 3
The Cartesian product set. Maps 4
Equivalence relations. Factoring a map through an equivalence relation
The natural numbers 15
The number system Z of integers 19
Some basic arithmetic facts about Z 22
A word on cardinal numbers 24

MONOIDS A N D GROUPS

1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
2

1

26

Monoids of transformations and abstract monoids 28
Groups of transformations and abstract groups 31
Isomorphism. Cayley's theorem 36
Generalized associativity. Commutativity 39
Submonoids and subgroups generated by a subset. Cyclic groups 42
Cycle decomposition of permutations 48
Orbits. Cosets of a subgroup 51
Congruences. Quotient monoids and groups 54
Homomorphisms 58
Subgroups of a homomorphic image. Two basic isomorphism theorems 64
Free objects. Generators and relations 67
Groups acting on sets 71
Sylow's theorems 79

RINGS

2.1
2.2
2.3
2.4
2.5

10

85

Definition and elementary properties
Types of rings 90
Matrix rings 92
Quaternions 98
Ideals' quotient rings 101

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xi

Contents

2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17

3

M O D U L E S OVER A P R I N C I P A L I D E A L D O M A I N

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11

4

Ideals and quotient rings for Z 103
Homomorphisms of rings. Basic theorems 106
Anti-isomorphisms 111
Field of fractions of a commutative domain 115
Polynomial rings 119
Some properties of polynomial rings and applications 127
Polynomial functions 134
Symmetric polynomials 138
Factorial monoids and rings 140
Principal ideal domains and Euclidean domains 147
Polynomial extensions of factorial domains 151
"Rings" (rings without unit) 155

Ring of endomorphisms of an abelian group 158
Left and right modules 163
Fundamental concepts and results 166
Free modules and matrices 170
Direct sums of modules 175
Finitely generated modules over a p.i.d. Preliminary results 179
Equivalence of matrices with entries in a p.i.d. 181
Structure theorem for finitely generated modules over a p.i.d. 187
Torsion modules' primary components' invariance theorem 189
Applications to abelian groups and to linear transformations 194
The ring of endomorphisms of a finitely generated module over a p.i.d. 204

G A L O I S T H E O R Y OF E Q U A T I O N S

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16

157

1

210

1

Preliminary results some old some new 213
Construction with straight-edge and compass 216
Splitting field of a polynomial 224
Multiple roots 229
The Galois group. The fundamental Galois pairing 234
Some results on finite groups 244
Galois' criterion for solvability by radicals 251
The Galois group as permutation group of the roots 256
The general equation of the nth degree 262
Equations with rational coefficients and symmetric group as
Galois group 267
Constructible regular rc-gons 271
Transcendence of e and n. The Lindemann-Weierstrass theorem 277
Finite fields 287
Special bases for finite dimensional extension fields 290
Traces and norms 296
Mod p reduction 301

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xii
5

Contents of Basic Algebra 1

REAL P O L Y N O M I A L EQUATIONS A N D INEQUALITIES

5.1
5.2
5.3
5.4
5.5
5.6

6

n

405

Definition and examples of associative algebras 406
Exterior algebras. Application to determinants 411
Regular matrix representations of associative algebras. Norms and traces 422
Change of base field. Transitivity of trace and norm 426
Non-associative algebras. Lie and Jordan algebras 430
Hurwitz' problem. Composition algebras 438
Frobenius' and Wedderburn's theorems on associative division algebras 451

LATTICES A N D B O O L E A N A L G E B R A S

8.1
8.2
8.3
8.4
8.5
8.6

342

Linear functions and bilinear forms 343
Alternate forms 349
Quadratic forms and symmetric bilinear forms 354
Basic concepts of orthogonal geometry 361
Witt's cancellation theorem 367
The theorem of Cartan-Dieudonne 371
Structure of the linear group GL (F) 375
Structure of orthogonal groups 382
Symplectic geometry. The symplectic group 391
Orders of orthogonal and symplectic groups over a finite field 398
Postscript on hermitian forms and unitary geometry 401

A L G E B R A S OVER A FIELD

7.1
7.2
7.3
7.4
7.5
7.6
7.7

8

316

METRIC VECTOR SPACES A N D THE CLASSICAL GROUPS

6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11

7

Ordered fields. Real closed fields 307
Sturm's theorem 311
Formalized Euclidean algorithm and Sturm's theorem
Elimination procedures. Resultants 322
Decision method for an algebraic curve 327
Generalized Sturm's theorem. Tarski's principle 335

306

455

Partially ordered sets and lattices 456
Distributivity and modularity 461
The theorem of Jordan-Holder-Dedekind 466
The lattice of subspaces of a vector space.
Fundamental theorem of projective geometry 468
Boolean algebras 474
The Mobius function of a partialy ordered set 480

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Preface

The most extensive changes in this edition occur in the segment of the book
devoted to commutative algebra, especially in Chapter 7, Commutative Ideal
Theory: General Theory and Noetherian Rings; Chapter 8, Field Theory; and
Chapter 9, Valuation Theory. In Chapter 7 we give an improved account of
integral dependence, highlighting relations between a ring and its integral extensions ("lying over," "going-up," and "going-down" theorems). Section 7.7,
Integrally Closed Domains, is new, as are three sections in Chapter 8: 8.13,
Transcendency Bases for Domains; 8.18, Tensor Products of Fields; and 8.19,
Free Composites of Fields. The latter two are taken from Volume III of our
Lectures in Abstract Algebra (D. Van N o s t r a n d 1964; Springer-Verlag, 1980).
The most notable addition to Chapter 9 is Krasner's lemma, used to give an
improved proof of a classical theorem of Kurschak's lemma (1913). We also
give an improved proof of the theorem on extensions of absolute values to a
finite dimensional extension of a field (Theorem 9.13) based on the concept of
composite of a field considered in the new section 8.18.
In Chapter 4, Basic Structure Theory of Rings, we give improved accounts

of the characterization of finite dimensional splitting fields of central simple
algebras and of the fact that the Brauer classes of central simple algebras over

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xiv

Preface

a given field constitute a set—a fact which is needed to define the Brauer group
Br(F). In the chapter on homological algebra (Chapter 6), we give an improved
proof of the existence of a projective resolution of a short exact sequence of
modules.
A number of new exercises have been added and some defective ones have
been deleted.
Some of the changes we have made were inspired by suggestions made by
our colleagues, Walter Feit, George Seligman, and Tsuneo Tamagawa. They,
as well as Ronnie Lee, Sidney Porter (a former graduate student), and the
Chinese translators of this book, Professors Cao Xi-hua and W a n g Jian-pan,
pointed out some errors in the first edition which are now corrected. We are
indeed grateful for their interest and their important inputs to the new edition.
O u r heartfelt thanks are due also to F. D. Jacobson, for reading the proofs of
this text and especially for updating the index.
January

1989

Nathan

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Jacobson


Preface to the First Edition

This volume is a text for a second course in algebra that presupposes an introductory course covering the type of material contained in the Introduction and
the first three or four chapters of Basic Algebra I. These chapters dealt with
the rudiments of set theory, group theory, rings, modules, especially modules
over a principal ideal domain, and Galois theory focused on the classical problems of solvability of equations by radicals and constructions with straight-edge
and compass.
Basic Algebra II contains a good deal more material than can be covered in
a year's course. Selection of chapters as well as setting limits within chapters
will be essential in designing a realistic program for a year. We briefly indicate
several alternatives for such a program: Chapter 1 with the addition of section
2.9 as a supplement to section 1.5, Chapters 3 and 4, Chapter 6 to section 6.11,
Chapter 7 to section 7.13, sections 8.1-8.3, 8.6, 8.12, Chapter 9 to section 9.13.
A slight modification of this program would be to trade off sections 4.6-4.8
for sections 5.1-5.5 and 5.9. F o r students who have had no Galois theory it
will be desirable to supplement section 8.3 with some of the material of Chapter 4 of Basic Algebra I. If an important objective of a course in algebra is an
understanding of the foundations of algebraic structures and the relation be-

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xvi

Preface to the First Edition

tween algebra and mathematical logic, then all of Chapter 2 should be included
in the course. This, of course, will necessitate thinning down other parts, e.g.,
homological algebra. There are many other possibilities for a one-year course
based on this text.
The material in each chapter is treated to a depth that permits the use of the
text also for specialized courses. F o r example, Chapters 3, 4, and 5 could constitute a one-semester course on representation theory of finite groups, and
Chapter 7 and parts of Chapters 8, 9, and 10 could be used for a one-semester
course in commutative algebras. Chapters 1, 3, and 6 could be used for an
introductory course in homological algebra.
Chapter 11 on real fields is somewhat isolated from the remainder of the
book. However, it constitutes a direct extension of Chapter 5 of Basic Algebra
I and includes a solution of Hilbert's problem on positive semi-definite rational
functions, based on a theorem of Tarski's that was proved in Chapter 5 of the
first volume. Chapter 11 also includes Pfister's beautiful theory of quadratic
forms that gives an answer to the question of the minimum number of squares
required to express a sum of squares of rational functions of n real variables
(see section 11.5).
Aside from its use as a text for a course, the book is designed for independent
reading by students possessing the background indicated. A great deal of material is included. However, we believe that nearly all of this is of interest to
mathematicians of diverse orientations and not just to specialists in algebra. We
have kept in mind a general audience also in seeking to reduce to a minimum
the technical terminology and in avoiding the creation of an overly elaborate
machinery before presenting the interesting results. Occasionally we have had
to pay a price for this in proofs that may appear a bit heavy to the specialist.
M a n y exercises have been included in the text. Some of these state interesting
additional results, accompanied with sketches of proofs. Relegation of these to
the exercises was motivated simply by the desire to reduce the size of the text
somewhat. The reader would be well advised to work a substantial number of
the exercises.
An extensive bibliography seemed inappropriate in a text of this type. In its

place we have listed at the end of each chapter one or two specialized texts in
which the reader can find extensive bibliographies on the subject of the chapter.
Occasionally, we have included in our short list of references one or two papers
of historical importance. N o n e of this has been done in a systematic or comprehensive manner.
Again it is a pleasure for me to acknowledge the assistance of many friends in
suggesting improvements of earlier versions of this text. I should mention first
the students whose perceptions detected flaws in the exposition and sometimes
suggested better proofs that they had seen elsewhere. Some of the students who

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xvii

Preface to the First Edition

have contributed in this way are Monica Barattieri, Ying Cheng, Daniel Corro,
William Ellis, Craig Huneke, and Kenneth McKenna. Valuable suggestions
have been communicated to me by Professors Kevin McCrimmon, James D.
Reid, Robert L. Wilson, and Daniel Zelinsky. I have received such suggestions
also from my colleagues Professors Walter Feit, George Seligman, and Tsuneo
Tamagawa. The arduous task of proofreading was largely taken over by Ying
Cheng, Professor Florence Jacobson, and James Reid. Florence Jacobson assisted in compiling the index. Finally we should mention the fine j o b of typing
that was done by Joyce Harry and D o n n a Belli. I am greatly indebted to all
of these individuals, and I take this opportunity to offer them my sincere thanks.
January

Nathan

1980

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Jacobson


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Introduction

In the Introduction to Basic Algebra I (abbreviated throughout as "BAI") we
gave an account of the set theoretic concepts that were needed for that volume.
These included the power set 0>(S) of a set S, the Cartesian product S x S of
two sets S and S , maps ( = functions), and equivalence relations. In the first
volume we generally gave preference to constructive arguments and avoided
transfinite methods altogether.
The results that are presented in this volume require more powerful tools,
particularly for the proofs of certain existence theorems. M a n y of these proofs
will be based on a result, called Zorn's lemma, whose usefulness for proving
such existence theorems was first noted by M a x Zorn. We shall require also
some results on the arithmetic of cardinal numbers. All of this fits into the
framework of the Zermelo-Fraenkel axiomatization of set theory, including
the axiom of choice (the so-called Z F C set theory). Two excellent texts that
can be used to fill in the details omitted in our discussion are P. R. Halmos'
Naive Set Theory and the more substantial Set Theory and the Continuum
Hypothesis by P. J. Cohen.
Classical mathematics deals almost exclusively with structures based on sets.
O n the other hand, category theory—which will be introduced in Chapter 1 —
1

1

2

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2


Introduction

2

deals with collections of sets, such as all groups, that need to be treated
differently from sets. Such collections are called classes. A brief indication of a
suitable foundation for category theory is given in the last section of this
Introduction.

0.1

ZORN'S L E M M A

We shall now formulate a maximum principle of set theory called Zorn's
lemma. We state this first for subsets of a given set. We recall that a set C of
subsets of a set S (that is, a subset of the power set £?{S)) is called a chain if C
is totally ordered by inclusion, that is, for any A,BeC either A a B or B cz A.
A set T of subsets of S is called inductive if the union [JA of any chain
C = {A^} cz T is a member of T. W e can now state
a

Z O R N ' S L E M M A (First formulation). Let T be a non-vacuous set of subsets
of a set S. Assume T is inductive. Then T contains a maximal element, that is,
there exists an MeT such that no AeT properly contains M.
There is another formulation of Zorn's lemma in terms of partially ordered
sets (BAI, p. 456). Let P, ^ be a partially ordered set. We call P, ^ (totally or
linearly) ordered if for every a,beP either a ^ b or b ^ a. We call P inductive if
every non-vacuous subset C of P that is (totally) ordered by ^ as defined in P
has a least upper bound in P, that is, there exists a u e P such that u ^ a for
every aeC and ifv^a
for every aeC then v ^ u. Then we have

Z O R N ' S L E M M A (Second formulation). Let P,^ be a partially ordered set
that is inductive. Then P contains maximal elements, that is, there exists meP
such that no aeP satisfies m < a.
It is easily seen that the two formulations of Zorn's lemma are equivalent, so
there is no harm in referring to either as "Zorn's lemma." It can be shown that
Zorn's lemma is equivalent to the

A X I O M O F C H O I C E . Let S be a set, ^(S. Then there exists a map f (a "choice function") of ^%S)* into S such that
f(A)eAfor
every Ae0>(S)*.
This is equivalent also to the following: If {A } is a set of non-vacuous sets
A , then the Cartesian product J][v4 # 0.
The statement that the axiom of choice implies Zorn's lemma can be proved
a

a

a

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0.2

3

Arithmetic of Cardinal Numbers

by an argument that was used by E. Zermelo to prove that every set can be
well ordered (see Halmos, pp. 62-65). A set S is well ordered by an order
relation ^ if every non-vacuous subset of S has a least element. The wellordering theorem is also equivalent to Zorn's lemma and to the axiom of
choice. We shall illustrate the use of Zorn's lemma in the next section.

0.2

A R I T H M E T I C OF C A R D I N A L N U M B E R S

Following Halmos, we shall first state the main results on cardinal arithmetic
without defining cardinal numbers. We say that the sets A and B have the
same cardinality and indicate this by \A\ = \B\ if there exists a bijective m a p of
A onto B. We write \A\ < \B\ if there is an injective m a p of A into B and
\A\ < \B\ if \A\ ^ \B\ and
\A\ ^ \B\. Using
these notations,
the
Schroder-Bernstein theorem (BAI, p. 25) can be stated as: \A\ ^ \B\ and
\B\ ^ \A\ if and only if \A\ = \B\. A set F is finite if \F\ = \N\ for some

N = { 0 , 1 , . . . , n — 1 } and A is countably infinite if \A\ = \co\ for co = {0,1,2,...}.
It follows from the axiom of choice that if A is infinite ( = not finite), then
\co\ < \A\. We also have Cantor's theorem that for any A, \A\ < \&(A)\.
We write C = AO B for sets A,B,C if C = A uB and A nB = 0. It is
and j i ? ^ ^ ^ ! , then \A 0B \
^\A 0B \.
Let
clear that if \A | ^\A \
C = F Oco where F is finite, say, F = { x , . . . , x _ i } where x # Xj for i ^ j .
Then the m a p of C into co such that x ~> i, k^k + n for keco is bijective.
Hence |C| = |co|. It follows from |co| ^ \A\ for any infinite A that if C = F 0 ^4,
then |C| = |^4|. F o r we can write A = DOB where \D\ = \co\. Then we have a
bijective m a p o f f u D onto D and we can extend this by the identity on B to
obtain a bijective m a p of C onto A.
We can use the preceding result and Zorn's lemma to prove the main result
on "addition of cardinals," which can be stated as: If A is infinite and
C = A u B where \B\ = \A\, then \C\ = \A\. This is clear if A is countable from
the decomposition co = {0,2,4,...} u {1,3,5,...}. It is clear also that the result
is equivalent to \A x 2| = \A\ if 2 = {0,1}, since \A x 2| = \A 0 B|. We proceed
to prove that \A x 2| = | ^ | for infinite A. Consider the set of pairs (XJ) where
X is an infinite subset of A a n d / i s a bijective m a p of X x 2 onto X. This set is
not vacuous, since A contains countably infinite subsets X and for such an X
we have bijective maps of 1 x 2 onto X. We order the set {(XJ)}
by
(XJ) < (X'J ) if X c X and / is an extension o f / It is clear that {(XJ)},
^
is an inductive partially ordered set. Hence, by Zorn's lemma, we have a
maximal (Y,g) in {(XJ)}. We claim that A — Y is finite. For, if ^4 — Y is
infinite, then this contains a countably infinite subset D, and gr can be extended
to a bijective m a p of (YOD) x 2 onto 7 0 D contrary to the maximality of

X

2

1

0

n

t

1

r

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x

2

t

2


4

Introduction

(Y,g). Thus A-Yis

finite.

Then

\A x 2| = |((Y x 2) u (A - Y)) x 2| = | Y x 2|
= |Y| = ^ | .
We can extend the last result slightly to
AOB

a)

= LB

if £ is infinite and \B\ 3* U|. This follows from
|B| < U u B | < B x 2 = Bl
The reader is undoubtedly familiar with the fact that \AxA\
countably infinite, which is obtained by enumerating co x co as

= \A\ if A is

(0,0), (0,1), (1,0), (0,2), (1,1), ( 2 , 0 ) , . . . .
More generally we have \A x ,4| = |^4| for any infinite A. The proof is similar to
the one for addition. We consider the set of pairs (X,f) where X is an infinite
subset of A and / is a bijective m a p of X x X onto X and we order the set
{(X,f)} as before. By Zorn's lemma, we have a maximal (Y,g) in this set. The
result we wish to prove will follow if \ Y\ = \A\. Hence suppose \ Y\ < \A\. Then
the relation A = Y 0(A— Y) and the result on addition imply that

\A\ = \A-Y|. Hence \A-Y\
> \ Y\. Then A-Y
contains a subset Z such that
| Z | = | Y|. Let W = Y u Z, so
= Y 0 Z and
x W is the disjoint union of
the sets Y x Y, Y x Z, Z x Y, and Z x Z. We have
| ( Y x Z ) u ( Z x Y ) u ( Z x Z)| = |Z x Z| = |Z|
Hence we can extend g to a bijective m a p of W x W onto FT. This contradicts
the maximality of (Y,g). Hence \ Y\ = \A\ and the proof is complete.
We also have the stronger result that if A ^ 0 and B is infinite and
|JB| > \A\, then
| 4 x B | = |B|.

(2)
This follows from

|S| < 14 x B\ < |B x B| =

0.3

ORDINAL AND CARDINAL NUMBERS

In axiomatic set theory no distinction is made between sets and elements. One
writes AeB for "the set A is a member of the set JB." This must be

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0.3

Ordinal and Cardinal Numbers

5

distinguished from A cz B, which reads "A is a subset of Br (In the texts on set
theory one finds A c= B for our A cz B and A cz B for our v4 ^ 5.) One defines
A cz B to mean that CeA => C e £ . One of the axioms of set theory is that
given an arbitrary set C of sets, there is a set that is the union of the sets
belonging to C, that is, for any set C there exists a set U such that A e U if and
only if there exists a 5 such that AeB and £ e C . In particular, for any set A
we can form the successor A =Au{A}
where {^4} is the set having the
single member A.
The process of forming successors gives a way of defining the set co (= N) of
n ,...
natural numbers. We define 0 = 0 , 1 = 0 = {0}, 2 = 1 , . . . , n +1 =
and we define co to be the union of the set of natural numbers n. The natural
number n and the set co are ordinal numbers in a sense that we shall now
define. First, we define a set S to be transitive if A e S and B e A => B e S. This is
equivalent to saying that every member of S is a subset. We can now give
+

+

D E F I N I T I O N 0.1.
transitive.

+

+

An ordinal is a set oc that is well ordered by e and is

The condition A e A is excluded by the axioms of set theory. We write A < B
for A e a, B e a if A e B or A = B. It is readily seen that every natural number n
is an ordinal and the set co of natural numbers is an ordinal. Also
co , (co ) , . . . are ordinals. The union of these sets is also an ordinal. This is
denoted as co + co or co x 2.
We shall now state without proofs some of the main properties of ordinal
numbers.
said to be similar if there
Two partially ordered sets S ^ and S <
exists an order-preserving bijective m a p of S onto S - The ordinals constitute
a set of representatives for the similarity classes of well-ordered sets. F o r we
have the following theorem: If S, ^ is well ordered, then there exists a unique
ordinal a and a unique bijective order-preserving m a p of S onto a. If a and
are ordinals, either a = p, a < /?, or f$ < a. An ordinal a is called a successor if
there exists an ordinal /? such that oc = P . Otherwise, a is called a Zimft ordinal.
Any non-vacuous set of ordinals has a least element.
+

+

+

a r e

l9

1

2i

2

x

2

+

D E F I N I T I O N 0.2. A cardinal number is an ordinal oc such that if P is any
ordinal such that the cardinality \P\ = |a|, then oc < /?.
A cardinal number is either finite or it is a limit ordinal. O n the other hand,
not every limit ordinal is a cardinal. F o r example co + co is not a cardinal. The
smallest infinite cardinal is co. Cardinals are often denoted by the Hebrew
letter "aleph" K with a subscript. In this notation one writes K for co.
0

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Introduction

6

Since any set can be well ordered, there exists a uniquely determined
cardinal a such that |a| = |S| for any given set S. We shall now call oc the
cardinal number or cardinality of S and indicate this by |5| = a. The results that

we obtained in the previous section yield the following formulas for
cardinalities
(3)

\AKJB\

= \B\

if B is infinite and \B\ ^ \A\. Here, unlike in equation (1), |S| denotes the
cardinal number of the set S. Similarly we have
(4)

'

\AxB\

= \B\

if A is not vacuous and \B\ is infinite and ^

0.4

SETS A N D

\A\.

CLASSES

There is an axiomatization of set theory different from the Z F system that
permits the discussion of collections of sets that may not themselves be sets.

This is a system of axioms that is called the Godel-Bernays (or GB) system.
The primitive objects in this system are "classes" and "sets" or more precisely
class variables and set variables together with a relation e. A characteristic
feature of this system is that classes that are members of other classes are sets,
that is, we have the axiom: Y e X => Y is a set.
Intuitively classes may be thought of as collections of sets that are defined
by certain properties. A part of the GB system is concerned with operations
that can be performed on classes, corresponding to combinations of properties.
A typical example is the intersection of classes, which is expressed by the
following: For all X and Y there exists a Z such that ueZ if and only if ueX
and UEY. We have given here the intuitive meaning of the axiom:
VXV Y 3 Z V u {ueZoueX
and ueY) where V is read "for all" and 3 is read
"there exists " Another example is that for every X there exists a Y such that
where
is the
ueY if and only if u£X i\JXlYMu
(ue Y <=> ~ueX),
negation of •••). Other class formations correspond to unions, etc. We refer to
Cohen's book for a discussion of the Z F and the GB systems and their
relations. We note here only that it can be shown that any theorem of Z F is a
theorem of GB and every theorem of GB that refers only to sets is a theorem
ofZF.
In the sequel we shall use classes in considering categories and in a few other
places where we encounter classes and then show that they are sets by showing
that they can be regarded as members of other classes. The familiar algebraic
structures (groups, rings, fields, modules, etc.) will always be understood to be
"small," that is, to be based on sets.

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0.4

7

Sets and Classes

REFERENCES

Paul R. Halmos, Naive Set Theory, Springer, New York, 1960.
Paul J. Cohen, Set Theory and the Continuum Hypothesis, Addison-Wesley, Reading,
Mass., 1966.

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1

Categories

In this chapter and the next one on universal algebra we consider two unifying
concepts that permit us to study simultaneously certain aspects of a large
number of mathematical structures. The concept we shall study in this chapter
is that of category, and the related notions of functor and natural
transformation. These were introduced in 1945 by Eilenberg and MacLane to
provide a precise meaning to the statement that certain isomorphisms are
"natural." A typical example is the natural isomorphism between a finitedimensional vector space V over a field and its double dual F**, the space of
linear functions on the space F * of linear functions on V. The isomorphism of
V onto F * * is the linear m a p associating with any vector xeV the evaluation

defined for a l l / e F * . To describe the "naturality" of this
function f^f(x)
isomorphism, Eilenberg and MacLane had to consider simultaneously all
finite-dimensional
vector spaces, the linear transformations between them, the
double duals of the spaces, and the corresponding linear transformations
between them. These considerations led to the concepts of categories and
functors as preliminaries to defining natural transformation. We shall discuss a
generalization of this example in detail in section 1.3.
The concept of a category is made up of two ingredients: a class of objects

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