Basic Structures of Modern Algebra

424 Pages·1993·13.466 MB·English

by  Yuri Bahturin (auth.)

Basic Structures of Modem Algebra
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 265
Basic Structures
of Modem Algebra
by
Yuri Bahturin
Department of Geometry and Topology,
Moscow State University,
Moscow, Russia
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4317-7 ISBN 978-94-017-0839-5 (eBook)
DOI 10.1007/978-94-017-0839-5
typeset by AMS-TEX
Printed on acid-free paper
All Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
TABLE OF CONTENTS
Preface .......... . vii
Recommendations to the reader . ix
Introduction. . . . 1
§ 1. Groups ....... . . 1
§ 2. Rings, fields . . . . . . 19
§ 3 Modules and representations . 32
Chapter 1. Commutative algebra . 51
§ 1. Algebraic and transcendental extensions 51
§ 2. Galois theory . . . . . . . . 60
§ 3. Affine rings . . . . . . . . . 69
§ 4. Modules over principal ideal rings 76
§ 5. Algebraic sets 86
§ 6. Normed fields . . . . . 95
Chapter 2. Groups . . . . . . 105
§ 1. Representations of groups 105
§ 2. Periodic groups .... 114
§ 3. Free groups and graphs 123
§ 4. Representation of groups by generators and relations 129
§ 5. Simple groups . . . 138
§ 6. Topological groups . 144
Chapter 3. Associative rings 157
§ 1. Radical . . . . . . 157
§ 2. Classical semisimple rings 165
§ 3. Structure of noetherian rings 169
§ 4. Central simple algebras 175
§ 5. Complete rings of fractions 190
Chapter 4. Lie algebras . . . . 199
§ 1. Linear Lie algebras . . . 199
§ 2. Universal enveloping algebra 208
§ 3. Magnus theory of free groups 215
§ 4. Lie algebras with triangular decomposition 224
§ 5. Lie algebras and Lie groups . . . . . . 235
v
vi TABLE OF CONTENTS
Chapter 5. Homological algebra 245
§ 1. Complexes of modules 245
§ 2. Cohomology of groups 254
§ 3. Splitting of the radical in a finite-dimensional algebra. 267
§ 4. Brauer group 273
§ 5. Hopf algebras . . . . . . . . . 280
Chapter 6. Algebraic groups . . . . . . 289
§ 1. Hopf algebras and algebraic groups 289
§ 2. Action of an algebraic group on a set 305
§ 3. Action of an algebraic group by linear operators 314
§ 4. Solvable groups . . . . . . 327
Chapter 7. Varieties of algebras ....... 335
§ 1. Universal algebras and varieties . . . . . 335
§ 2. Finite basis problem for identities in groups 346
§ 3. PI-algebras . . . . . . . . . . . . 353
§ 4. Central polynomials for matrix algebras 364
Set-theoretic supplement 375
References . 397
Symbol index 401
Subject index 405
PREFACE
This book has developed from a series of lectures which were given by the
author in mechanics-mathematics department of the Moscow State University.
In 1981 the course "Additional chapters in algebra" replaced the course "Gen
eral algebra" which was founded by A.G. Kurosh (1908-1971), professor and
head of the department of higher algebra for a period of several decades. The
material of this course formed the basis of A. G. Kurosh's well-known book
"Lectures on general algebra" (Moscow,1962; 2-nd edition: Moscow, Nauka,
1973) and the book "General algebra. Lectures of 1969-1970." (Moscow,
Nauka, 1974). Another book based on the course, "Elements of general al
gebra" (M.: Nauka, 1983) was published by L.A. Skorniakov, professor, now
deceased, in the same department. It should be noted that A.G. Kurosh was
not only the lecturer for the course "General algebra" but he was also the
recognized leader of the scientific school of the same name. It is difficult to
determine the limits of this school; however, the "Lectures ... " of 1962 men
tioned above contain some material which exceed these limits. Eventually this
effect intensified: the lectures of the course were given by many well-known
scientists, and some of them see themselves as "general algebraists". Each
lecturer brought significant originality not only in presentation of the material
but in the substance of the course. Therefore not all material which is now
accepted as necessary for algebraic students fits within the scope of general
algebra.
There is another side to this problem. The fact is that some chapters from
"general algebra" are so similar (see the second of the books cited by Kurosh)
that it seems no longer advisable to study even just the basics of all these
chapters for all algebraists.
These arguments lead us to change the title of this course of which the
main goal is to introduce beginners to the concepts, results and problems of
contemporary algebra assuming knowledge of the standard theory of linear
algebra and vector spaces.
The idea of this book is not only to introduce the various concepts but also
to show how they work and relate. Though the presentation of the material
has a somewhat of a "chunky" character: a chunk from the theory of fields,
Vll
viii PREFACE
a chunk from groups, rings, Lie algebras, etc .. There are, on the other hand,
themes which "roam" from one chapter to another. One of the more significant
of these themes is the one of periodic groups which started with the classical
work on the theory of groups by W. Burnside. Another organizing principle
is construction the bridges (links): groups - graphs, groups - rings - Lie
algebras, fields - skew fields - varieties of algebras, etc.. The reader should
judge for himself the value of this result.
It is pleasure for me to express my sincere gratitude to the translator D. Zvire
naite and to the editor Prof. M. Hazewinkel who did an excelent job and thus
saved me from a certain amount of criticism to follow.
RECOMMENDATIONS TO THE READER
The author of this book tried to make the account self contained as far as
this is possible in a text which is based on a general university courses for
the first two academic years. Practically, all the concepts of abstract algebra
used in this course are collected in the introduction which also contains some
additional background material. References to this part of the book in other
chapters contain the word Introduction: a reference Introduction m. n means
that it concerns subsection n of section m from the Introduction. Various set
theoretic concepts and theorems, as well as a number of results from topology
and the theory of smooth manifolds are contained in a set-theoretic appendix
at the end of the book. References to this section are indicated by Set
theoretic Supplement n, where n is a number of a subsection in the appendix.
References to a certain subsection in a chapter are denoted by m. n. k, where
m is the number of a chapter, n the number of a section, k the number of
a subsection. There also occur references of the form § m. n, where m is
the number of a chapter, n the number of a section. A reference inside a
chapter does not mention that chapter and a reference inside a section does
not mention that section.
The subject and symbol indexes indicate where a concept or notation was
first introduced.
The end of a proof is marked by the sign "0". If no proof is given the
same sign occurs at the end of the statement. Some simple extra statements
are indicated by circles: o. Supplementary material is marked by stars
0 •••
* ... *.
ix
INTRODUCTION
The purpose of this introduction is to explain in a condensed form the main
definitions and results in group, ring and field theory that are usually presented
in elementary standard courses of a study in physics or mathematics.
§ 1. Groups
1.1. Definitions. A set G with a binary associative operation (g, h) g h
1--+
is called a group if it contains a unit element, i.e. an element e such that
= =
ge eg g for any g E G, and if for each g E G there exists an inverse g-l,
= =
i.e. gg-l g-lg e.
If the operation is written multiplicatively, then the unit element is called an
identity and denoted as 1.
When an additive description of the operation is used, i.e. (g, h) - g + h,
the unit element is called the zero and is denoted as 0, the inverse element
= =
to g is called the negative and is denoted as -g; thus g + 0 0 + g g and
g + (-g) = (-g) + g = O.
The unit element is unique; unique also is the inverse element for any given
g.
A non-empty subset H of a group G is called a subgroup if for any hI, h2 E
H one has hlh2 E H, hII, hZI E H.
o A subset H of a group G is a subgroup iff H is a group relative to the same
operation as G.
0
A proper subgroup is any subgroup not equal to the whole group.
The trivial subgroup is the subgroup consisting only of the unit element.
Two groups are called isomorphic if there exists a bijection 4>: GI - G2
such that the result of the operation on the images of any two elements from
GI coincides with the image of the result of the operation on these elements.
We write G I '::! G2.
For instance, if the operation is written multiplicatively in GI and G2, x, Y E
GI, then the statement has the form
1