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

219 Pages·3.125 MB·English
by  S. T Hu
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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 Vii 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

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