Noncommutative Noetherian Rings

This is a reprinted edition of a work that was considered the definitive account in the subject area upon its initial publication by J. Wiley & Sons in 1987. It presents, within a wider context, a comprehensive account of noncommutative Noetherian rings. The author covers the major developments from the 1950s, stemming from Goldie's theorem and onward, including applications to group rings, enveloping algebras of Lie algebras, PI rings, differential operators, and localization theory. The book is not restricted to Noetherian rings, but discusses wider classes of rings where the methods apply more generally. In the current edition, some errors were corrected, a number of arguments have been expanded, and the references were brought up to date. This reprinted edition will continue to be a valuable and stimulating work for readers interested in ring theory and its applications to other areas of mathematics.

This introduction to noncommutative noetherian rings is intended to be accessible to anyone with a basic background in abstract algebra. It can be used as a second-year graduate text, or as a self-contained reference. Extensive explanatory discussion is given, and exercises are integrated throughout. This edition incorporates substantial revisions, particularly in the first third of the book, where the presentation has been changed to increase accessibility and topicality. New material includes the basic types of quantum groups, which then serve as test cases for the theory developed.

Introduces and applies the standard techniques in the area (ring of fractions, bimodules, Krull dimension, linked prime ideals).

Commutative algebra is a rapidly growing subject that is developing in many different directions. This volume presents several of the most recent results from various areas related to both Noetherian and non-Noetherian commutative algebra. This volume contains a collection of invited survey articles by some of the leading experts in the field. The authors of these chapters have been carefully selected for their important contributions to an area of commutative-algebraic research. Some topics presented in the volume include: generalizations of cyclic modules, zero divisor graphs, class semigroups, forcing algebras, syzygy bundles, tight closure, Gorenstein dimensions, tensor products of algebras over fields, as well as many others. This book is intended for researchers and graduate students interested in studying the many topics related to commutative algebra.

This volume collects some of the survey lectures delivered at the Micro program on Noncommutative Rings held at MSRI, July 10-21, 1989. While the program was concerned with recent advances in ring theory, it also had as an important component lectures on related areas of mathematics where ring theory might be expected to have an impact. Thus, there are lectures of S. P. Smith on quantum groups and Marc Ri effel on algebraic aspects of quantum field theory. Martin Lorenz and Don ald Passman consider in their lectures various aspects of crossed products: homological and K-theoretic to group actions. Kenneth Brown presents the "modern" theory of Noetherian rings and localization. These contributions as well as the others not presented here show that ring theory remains a vigorous and useful area. The planning and organization of the program were done by the under signed and the late Robert Warfield. His illness prevented his attendance at the meeting. It is to him we dedicate this volume. The organizers wish to extend their thanks to Irving Kaplansky, Director of MSRI, and the staff for all of their efforts in making this conference such a success. Susan Montgomery Lance Small vii NONCOMMUTATIVE RINGS TABLE OF CONTENTS PREFACE. . . . . . . . . . . . . . . . . . . . . . .. . . . Vll . . . . . . . . . . K. A. Brown THE REPRESENTATION THEORY OF NOETHERIAN RINGS 1 A. Joseph SOME RING THEORETIC TECHNIQUES AND OPEN PROBLEMS IN ENVELOPING ALGEBRAS. . . . . . . . . . . 27 . . . M. Lorenz CROSSED PRODUCTS: CHARACTERS, CYCLIC HOMOLOGY, AND GROTHENDIECK GROUPS . . . . . . . . . . . . . . . 69 . . . . . .

". T. Stafford -- The Goldie rank of a module " . R. Farkas -- Noetherian group rings: An exercise in creating folklore and intuition " . C. Jantzen -- Primitive ideals in the enveloping algebra of a semisimple Lie algebra " . J. Enright -- Representation theory of semisimple Lie algebras " .-E. Björk -- Filtered Noetherian rings " . Rentschler -- Primitive ideals in enveloping algebras.

Providing an elementary introduction to noncommutative rings and algebras, this textbook begins with the classical theory of finite dimensional algebras. Only after this, modules, vector spaces over division rings, and tensor products are introduced and studied. This is followed by Jacobson's structure theory of rings. The final chapters treat free algebras, polynomial identities, and rings of quotients. Many of the results are not presented in their full generality. Rather, the emphasis is on clarity of exposition and simplicity of the proofs, with several being different from those in other texts on the subject. Prerequisites are kept to a minimum, and new concepts are introduced gradually and are carefully motivated. Introduction to Noncommutative Algebra is therefore accessible to a wide mathematical audience. It is, however, primarily intended for beginning graduate and advanced undergraduate students encountering noncommutative algebra for the first time.