Elements of modern algebra (Holden-Day series in mathematics)

219 Pages·3.125 MB·English

by  S. T Hu

Elements of
MODERN ALGEBRA
HOLDEN-DAY SERIES IN MATHEMATICS
Earl A. Coddzngton and Andrew M. Gleason, Editors
G. HOCHSCHILD, The Structure of Lie Groups
SZE-TsEN Hu, Elements of General Topology
SZE-TSEN Hu, Elements of Modern Algebra
MCCOART, OLIPHANT, and SCHEERER, Elementary Analysis
Elements of
MODERN ALGEBRA
Sze-Tsen Hu
Department of Mathematics
University of California, Los Angeles
HOLDEN-DAY
San Francisco, Cambridge, London, Amsterdam
© Copyright 1965 by Holden-Day, Inc., 50o Sansome Street, San Francisco,
California All rights reserved. No part of this book may be reproduced in any form,
by mimeograph or any other means, without permission in writing from the publisher.
Library of Congress Catalog Card Number 65-21823
Printed in the United States of America
PREFACE
Abstract algebra is now included in the undergraduate curricula of
most universities. It has become an essential part of the training of
mathematicians. The present book is designed as a text for a one-
semester or two-quarter course of the subject for upper division under-
graduates as well as first-year graduate students. Its aim is to provide a
systematic exposition of the essentials of this subject in a desirably leisurely
fashion, to students who have reached at least the level of mathematical
maturity following two or three years of sound undergraduate mathe-
matics study. Apart from the arithmetic of real numbers, no specific
mathematical knowledge is required
The first four chapters can be used as a text for a one-quarter course,
or, when slightly supplemented, a one-semester course in group theory.
Here, emphasis is placed on Abelian groups instead of finite permutation
groups. In addition to the more or less standard materials of group
theory, we give elementary accounts of exact sequences, homology
groups, tensor products and groups of homomorphisms.
The fifth chapter gives a condensed study of rings, integral domains
and fields. The sixth chapter presents an elementary theory of modules
and algebras leading to the construction of the tensor algebra, the exterior
algebra, and the symmetric algebra of a given module. In the final
chapter, we introduce the student to the relatively new concept of cate-
gories and functors which has become essential in many branches of
mathematics.
For pedagogical reasons, certain usual topics of abstract algebra are
deliberately omitted, most notably linear algebra and Galois theory.
Linear algebra is omitted here because it is now often taught either as a
separate course or as a part of a two-year calculus series. On the other
hand, Galois theory is omitted since it seems to the author that, in view
of its deepness, it belongs to the last quarter of a year course instead of the
first two.
As a rule, repetition is not avoided. On the contrary, we deliberately
repeat important formulations on different objects as close as possible.
For example, the central idea of a universal algebra by means of a commu-
a
vi Preface
tative triangle is repeated in the definitions of free semigroups, free groups,
free Abelian groups, free modules, tensor products, tensor algebras,
exterior algebras, and symmetric algebras. In an elementary text such
as this, repetition of fundamental concepts and basic constructions in-
creases the confidence and mastery of the student.
The exercises at the end of each section are carefully chosen so that
the good student may have sufficient challenge to participate further in
the development of the theory while the other students are enjoying the
easy detailed expositions in the text.
The bibliography at the end of the book lists reference books of
various levels for further studies as well as for more examples and exercises.
A few references to this bibliography are cited in the text by names and
numbers enclosed in brackets. Cross references are given in the form
(IV, 5.1), where IV stands for Chapter IV and 5.1 for the numbering of
the statement in the chapter.
A list of special symbols and abbreviations used in this book is given
immediately after the Table of Contents. Certain deviations from
standard set-theoretic notations have been adopted in the text; namely,
is used to denote the empty set and A\B the set-theoretic difference
usually denoted by A-B. We have used the symbol I I to indicate the end
of a proof and the abbreviation if for the phrase "if and only if."
It is a great pleasure to acknowledge the invaluable assistance the
author received in the form of financial support from the Air Force Office
of Scientific Research during the years since 1957 while the present book
was gradually developed as various lecture notes. Finally, the author
wishes to thank the publisher and the printer for their courtesy and
cooperation.
Sze- Tsen Hu
University of California
Los Angeles, Calif.
TABLE OF CONTENTS
Special Symbols and Abbreviations
Chapter I: SETS, FUNCTIONS AND RELATIONS
1. Sets . . . . . . . . . . . . . . 1
2. Functions . . . . . . . . . . . . 6
3. Cartesian products . . . . . . . . . . 11
4. Relations . . . . . . . . . . . . 14
Chapter II: SEMIGROUPS
1. Binary operations . . . . . . . . . . 18
2. Definition of a semigroup . . . . . . . . 22
3. Homomorphisms . . . . . . . . . . . 26
4. Free semigroups . . . . . . . . . . . 30
Chapter III: GROUPS
1. Definition of a group . . . . . . . . . 36
2. Subgroups . . . . . . . . . . . . 39
3. Homomorphisms . . . . . . . . . . . 43
4. Quotient groups . . . . . . . . . . . 48
5. Finite groups . . . . . . . . . . . 55
6. Direct products . . . . . . . . . . . 59
7. Free groups . . . . . . . . . . . . 65
8. Exact sequences . . . . . . . . . . . 68
Chapter IV: ABELIAN GROUPS
1. Generalities . . . . . . . . . . . . 76
2. Free Abelian groups . . . . . . . . . . 80
3. Decomposition of cyclic groups . . . . . . . 85
4. Finitely generated Abelian groups . . . . . . 88
5. Semi-exact sequences . . . . . . . . . . 96
6. Tensor products . . . . . . . . . . . 99
7. Group of homomorphisms . . . . . . . . 109
Chapter V: RINGS, INTEGRAL DOMAINS AND FIELDS
1. Definitions and examples . . . . . . . . . 114
2. Subrings and ideals . . . . . . . . . . 119
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viii Table of Contents
3. Homomorphisms . . . . . . . . . 123
4. Characteristic . . . . . . . . . . . 128
5. Fields of quotients . . . . . . . . . . 131
6. Polynomial rings . . . . . . . . . . . 135
7. Factorization . . . . . . . . . . 139
Chapter VI: MODULES, VECTOR SPAC1'S AND
ALGEBRAS
1. Definitions and examples . . . . . . . 145
2. Submodules and subalgebras . . . . . . . 149
3. Homomorphisms . . . . . . . . . . . 153
4. Free modules . . . . . . . . . . . . 158
5. Tensor products . . . . . . . . . . . 163
6. Graded modules . . . . . . . . . . . 168
7. Graded algebras . . . . . . . . . . 173
8. Tensor algebras . . . . . . . . . . . 178
9. Exterior algebras . . . . . . . . . . 181
10. Symmetric algebras . . . . . . . . . . 185
Chapter VII: CATEGORIES AND FUNCTORS
1. Semigroupoids . . . . . . . . . . . 189
2. Categories . . . . . . . . . . . . 192
3. Functors . . . . . . . . . . . . . 195
4. Transformations of functors . . . . . . . . 198
Bibliography 201
Index 203
SPECIAL SYMBOLS AND
ABBREVIATIONS
implies
is implied by
i end of proof
II
if and only if
set such that
{ I }
E is a member of
is not a member of
empty set
C is contained in
contains
U union
n intersection
set-theoretic difference
I closed unit interval
Cx(A) complement of A with respect to X
f :X -+ Y function f from X to Y
f (A) image of the set A under f
f -1(B) inverse image of the set B
f o g the composition of f and g
f I A the restriction of f on A
a c t b a is Gt-related to b
a , b
a is equivalent to b
X/ quotient set of X over
X ,: Y X is isomorphic to Y
X/A quotient group, etc., of X over A
A Q+ B directed sum of A and B
A p B tensor product of A and B
A O B tensor product over R
R
ER (M) exterior algebra of M over R
SR (M) symmetric algebra of M over R
TR (M) tensor algebra of M over R
ix
x Specaal symbols and abbreviations
Coim coimage
Coker cokernel
deg degree
dim dimension
Horn group of homomorphisms
Im image
Ker kernel