A First Course in Noncommutative Rings

by T.Y. Lam

Overview

One of my favorite graduate courses at Berkeley is Math 251, a one-semester course in ring theory offered to second-year level graduate students. I taught this course in the Fall of 1983, and more recently in the Spring of 1990, both times focusing on the theory of noncommutative rings. This book is an outgrowth of my lectures in these two courses, and is intended for use by instructors and graduate students in a similar one-semester course in basic ring theory. Ring theory is a subject of central importance in algebra. Historically, some of the major discoveries in ring theory have helped shape the course of development of modern abstract algebra. Today, ring theory is a fer­ tile meeting ground for group theory (group rings), representation theory (modules), functional analysis (operator algebras), Lie theory (enveloping algebras), algebraic geometry (finitely generated algebras, differential op­ erators, invariant theory), arithmetic (orders, Brauer groups), universal algebra (varieties of rings), and homological algebra (cohomology of rings, projective modules, Grothendieck and higher K-groups). In view of these basic connections between ring theory and other branches of mathemat­ ics, it is perhaps no exaggeration to say that a course in ring theory is an indispensable part of the education for any fledgling algebraist. The purpose of my lectures was to give a general introduction to the theory of rings, building on what the students have learned from a stan­ dard first-year graduate course in abstract algebra.

Product Details

ISBN-13:	9781468404081
Publisher:	Springer New York
Publication date:	03/04/2012
Series:	Graduate Texts in Mathematics Series , #131
Edition description:	Softcover reprint of the original 1st ed. 1991
Pages:	397
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1. Wedderburn-Artin Theory.- §1. Basic terminology and examples.- §2. Semisimplicity.- §3. Structure of semisimple rings.- 2. Jacobson Radical Theory.- §4. The Jacobson radical.- §5. Jacobson radical under change of rings.- §6. Group rings and the J-semisimplicity problem.- 3. Introduction to Representation Theory.- §7. Modules over finite-dimensional algebras.- §8. Representations of groups.- §9. Linear groups.- 4. Prime and Primitive Rings.- §10. The prime radical; prime and semiprime rings.- §11. Structure of primitive rings; the Density Theorem.- §12. Subdirect products and commutativity theorems.- 5. Introduction to Division Rings.- §13. Division rings.- §14. Some classical constructions.- §15. Tensor products and maximal subfields.- §16. Polynomials over division rings.- 6. Ordered Structures in Rings.- §17. Orderings and preorderings in rings.- §18. Ordered division rings.- 7. Local Rings, Semilocal Rings, and Idempotents.- §19. Local rings.- §20. Semilocal rings.- §21. The theory of idempotents.- §22. Central idempotents and block decompositions.- 8. Perfect and Semiperfect Rings.- §23. Perfect and semiperfect rings.- §24. Homological characterizations of perfect and semiperfect rings.- §25. Principal indecomposables and basic rings.- References.- Name Index.

Editorial Reviews

From the reviews of the second edition:

MATHEMATICAL REVIEWS

"This is a textbook for graduate students who have had an introduction to abstract algebra and now wish to study noncummutative rig theory…there is a feeling that each topic is presented with specific goals in mind and that the most efficient path is taken to achieve these goals. The author received the Steele prize for mathematical exposition in 1982; the exposition of this text is also award-wining caliber. Although there are many books in print that deal with various aspects of ring theory, this book is distinguished by its quality and level of presentation and by its selection of material….This book will surely be the standard textbook for many years to come. The reviewer eagerly awaits a promised follow-up volume for a second course in noncummutative ring theory."

"Ten years ago, the first edition ... of this book appeared. It is quite rare that a book can become a classic in such a short time, but this did happen for this excellent book. Of course minor changes were made for the second edition; new exercises and an appendix on uniserial modules were added. Every part of the text was written with love and care. The explanations are very well done, useful examples help to understand the material ... ." (G. Pilz, Internationale Mathematische Nachrichten, Issue 196, 2004)

"The present book is a radical update. For the second edition the text was retyped, some proofs were rewritten and improvements in exposition have also taken place. ... It is well-written and consists of eight chapters. ... There is a very good reference section for further study and a name index consisting of four pages of closely-packed names. ... As always the standard of print and presentation by Springer is exemplary." (Brian Denton, The Mathematical Gazette, Vol. 86 (505), 2002)

From the Publisher