文档介绍：Basic ~l~ebra/l
Second Edition
NATHAN~OBSON
VALE UNIVERSIW
W. H. FREEMAN PANY
New York
Library of Congress Cstalogiog in Publiation Data
Jacobson, Nathan, 1910-
Basic algebra.
Includes index.
1. Algebra. I. Title.
1985 8425836
ISBN 0-7167-1480-9 (v. I)
Copyright @ 1985 by . Freeman pany
No part 01 this book may be reproduced
by any mechanical, photographic, or electronic process,
or in the lorn of a phonographic recording.
nor may it be stored in a retrieval system, transmitted,
or otherwise copied lor public or private use,
without written permission hom the publisher.
Printed in the United States of America
Contents
Preface xi
Preface to the First Edition xiii
INTRODUCTION: CONCEPTS FROM SET THEORY.
THE INTEGERS 1
The power set of a set 3
The Cartesian product set. Maps 4
Equivalence relations. Factoring a map through an
equivalence relation 10
The natural numbers 15
The number system E of integers 19
Some basic arithmetic facts about E 22
A word on cardinal numbers 24
1 MONOIDS AND GROUPS 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 position 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 tbeorems 61
Free objects. Gent-ators and relations 67
Groups acting on sets 71
Sylow's theorems 79
2 RINGS 85
Definition and elementary properties 86
Types of rings 90
Matrix rings 92
Quaternions 98
Ideals, quotient rings 101
Ideals and quotient rings for Z 103
Homomorphisms of rings. Basic theorems 106
An