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INTRODUCTORY LECTURES ON
RINGS AND MODULES

John A. Beachy

London Mathematical Society Student Texts 47
Cambridge University Press
April, 1999


Hardback Edition: $162.00, ISBN 0-521-64340-6, ISBN-13:9780521643405
Paperback Edition: $58.00, ISBN 0-521-64407-0, ISBN-13:9780521644075

Cambridge University Press,
Edinburgh Building,
Shaftesbury Road,
Cambridge CB2 2RU

Information on the WEB:

Detailed information about the book from Cambridge University Press
Cambridge University Press


INTRODUCTION

The focus of this book is the study of the noncommutative aspects of rings and modules, and the style will make it accessible to anyone with a background in basic abstract algebra. Features of interest include an early introduction of projective and injective modules; a module theoretic approach to the Jacobson radical and the Artin-Wedderburn theorem; the use of Baer's criterion for injectivity to prove the structure theorem for finitely generated modules over a principal ideal domain; and applications of the general theory to the representation theory of finite groups. Optional material includes a section on modules over the Weyl algebras and a section on Goldie's theorem. When compared to other more encyclopedic texts, the sharp focus of this book accommodates students meeting this material for the first time. It can be used as a first-year graduate text or as a reference for advanced undergraduates.


THIS SITE

includes the following:

For further information, please contact

John Beachy,
Dept. of Mathematical Sciences,
Northern Illinois University,
DeKalb, Illinois 60115
TEL: 815 / 753-6753

This site was inaugurated in September, 1998, and last modified on August 14, 2000.
This page has been accessed 102,359 times since 9/98.

email: jbeachy@niu.edu | Author's homepage


TABLE OF CONTENTS

PREFACE

CHAPTER 1: RINGS (62 pages)

1.1 Basic definitions and examples
1.2 Ring homomorphisms
1.3 Localization of integral domains
1.4 Unique factorization
1.5 *Additional noncommutative examples

CHAPTER 2: MODULES (74 pages)

2.1 Basic definitions and examples
2.2 Direct sums and products
2.3 Semisimple modules
2.4 Chain conditions
2.5 Modules with finite length
2.6 Tensor products
2.7 Modules over principal ideal domains
2.8 *Modules over the Weyl algebras

CHAPTER 3: STRUCTURE OF NONCOMMUTATIVE RINGS (34 pages)

3.1 Prime and primitive ideals
3.2 The Jacobson radical
3.3 Semisimple Artinian rings
3.4 *Orders in simple Artinian rings

CHAPTER 4: REPRESENTATIONS OF FINITE GROUPS (36 pages)

4.1 Introduction to group representations
4.2 Introduction to group characters
4.3 Character tables and orthogonality relations

APPENDIX (22 pages)

Review of vector spaces | Zorn's lemma | Matrices over commutative rings | Eigenvalues and characteristic polynomials | Noncommutative quotient rings | The ring of algebraic integers

BIBLIOGRAPHY | LIST OF SYMBOLS | INDEX

238 pages

* Sections that may be omitted from the development


PREFACE

This set of lecture notes is focused on the noncommutative aspects of the study of rings and modules. It is intended to complement the book Steps in Commutative Algebra, by R. Y. Sharp, which provides excellent coverage of the commutative theory. It is also intended to provide the necessary background for the book An Introduction to Noncommutative Noetherian Rings, by K. R. Goodearl and R. B. Warfield.

The core of the first three chapters is based on my lecture notes from the second semester of a graduate algebra sequence that I have taught at Northern Illinois University. I have added additional examples, in the hope of making the material accessible to advanced undergraduate students. To provide some variety in the examples, there is a short section on modules over the Weyl algebras. This section is marked with an asterisk, as it can be omitted without causing difficulties in the presentation. (The same is true of Section 1.5 and Section 3.4.) Chapter 4 provides an introduction to the representation theory of finite groups. Its goal is to lead the reader into an area in which there has been a very successful interaction between ring theory and group theory.

Certain books are most useful as a reference, while others are less encyclopedic in nature, but may be an easier place to learn the material for the first time. It is my hope that students will find these notes to be accessible, and a useful source from which to learn the basic material. I have included only as much material as I have felt it is reasonable to try to cover in one semester. The role of an encyclopedic text is played by any one of the standard texts by Jacobson, Hungerford, and Lang. My personal choice for a reference is Basic Algebra by N. Jacobson.

There are many possible directions for subsequent work. To study noncommutative rings the reader might choose one of the following books: An Introduction to Noncommutative Noetherian Rings, by K. R. Goodearl and R. B. Warfield, A First Course in Noncommutative Rings, by T. Y. Lam, and A Course in Ring Theory, by D. S. Passman. After finishing Chapter 4 of this text, the reader should have the necessary background to study Representations and Characters of Finite Groups, by M. J. Collins. Another possibility is to study A Primer of Algebraic D-Modules, by S. C. Coutinho.

I expect the reader to have had prior experience with algebra, either at the advanced undergraduate level, or in a graduate level course on Galois theory and the structure of groups. Virtually all of the prerequisite material can be found in undergraduate books at the level of Herstein's Abstract Algebra. For the sake of completeness, two definitions will be given at this point. A group is a nonempty set G together with a binary operation . on G such that the following conditions hold: (i) for all a,b,c in G, we have a.(b.c) = (a.b).c; (ii) there exists 1 in G such that 1.a = a and a.1 = a for all a in G; (iii) for each a in G there exists an element a-1 in G such that a.a-1 = 1 and a-1.a = 1. The group G is said to be abelian if a.b=b.a for all a,b in G, and in this case the symbol . for the operation on G is usually replaced by a + symbol, and the identity element is denoted by the symbol 0 rather than by the symbol 1. The definition of an abelian group is fundamental, since the objects of study in the text (rings and modules) are constructed by endowing an abelian group with additional structure.

I sincerely hope that the reader's prior experience with algebra has included the construction of examples. Good examples provide the foundation for understanding this material. I have included a variety of them, but it is best if additional ones are constructed by the reader. A good example illustrates the key ideas of a definition or theorem, but is not so complicated as to obscure the important points. Each definition should have several associated examples that will help in understanding and remembering the conditions of the definition. It is helpful to include some that do not satisfy the stated conditions.


SUPPLEMENT

From the preface to the supplement:

The text grew out of the first-year graduate course that I have taught at Northern Illinois University. In keeping with the noncommutative focus of the text, I included a chapter on group representations in place of the chapter on commutative rings that is in our course syllabus. Since a course at that level should lay some foundation for later work in commutative algebra, I felt that I should make my class notes available as a supplement to the text.

At this point, the supplement is approximately 30 pages long and includes the following sections.

Chapter 5: Commutative rings

5.1 Primary decomposition
5.2 Noetherian rings
5.3 Dedekind domains
5.4 Integral extensions
I have labeled it a preliminary version, because the notes refer to other sources for the proofs of the Hilbert nullstellensatz and the principal ideal theorem (among other results). I also intend to add further problems. Please check back later for a revised version.

Click here for the version posted in May, 1999.


CLASS NOTES

These notes are intended to help students who are working through the text. The notes include some background material, along with supplementary problems (with solutions). At some point in the future I hope to have a complete manuscript available on the web, with many more solved problems, and extensive comments on the text.

This part of my site is still "under construction".
Click here for the parts that are available now. (The individual files are in pdf format.)


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