TABLE OF CONTENTS
PREFACE
CHAPTER 1: RINGS | Basic definitions and examples |Ring homomorphisms | Localization of integral domains | Unique factorization | *Additional noncommutative examples || 62 pages
CHAPTER 2: MODULES | Basic definitions and examples | Direct sums and products | Semisimple modules | Chain conditions | Modules with finite length | Tensor products | Modules over principal ideal domains | *Modules over the Weyl algebras || 74 pages
CHAPTER 3: STRUCTURE OF NONCOMMUTATIVE RINGS | Prime and primitive ideals | The Jacobson radical | Semisimple Artinian rings | *Orders in simple Artinian rings || 34 pages
CHAPTER 4: REPRESENTATIONS OF FINITE GROUPS | Introduction to group representations | Introduction to group characters | Character tables and orthogonality relations || 36 pages
APPENDIX | Review of vector spaces | Zorn's lemma | Matrices over commutative rings | Eigenvalues and characteristic polynomials | Noncommutative quotient rings | The ring of algebraic integers || 22 pages
BIBLIOGRAPHY | LIST OF SYMBOLS | INDEX
238 pages
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.
From the 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.