Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0644.16008
McConnell, J.C.; Robson, J.C.
(Small, L.W.)
Noncommutative Noetherian rings. With the cooperation of L. W. Small. (English)
[B] Pure and Applied Mathematics. A Wiley-Interscience Publication. Chichester etc.: John Wiley \& Sons. XV, 596 p.; \sterling 65.00 (1987). ISBN 0-471-91550-5

Noncommutative Noetherian Ring Theory emerged as a discipline in its own right with the publication of Goldie's Theorems for prime and semiprime rings in 1958-60. Goldie's Theorem occupies the same place in the noetherian theory as the Artin-Wedderburn Theorem in the artinian theory. Indeed, Goldie's Theorem provides an important link between noetherian and artinian methods. In the thirty years since 1958 the subject has developed both as an intrinsically interesting branch of algebra and as a tool in other branches of algebra. Until the publication of this book there has been no attempt to provide an overview of, and a general reference for, the most important developments in the theory. The present authors have set out to fill this gap and have succeeded admirably. The book is written in a style that makes it easy to read and use: the authors aim for clarity of expression rather than for the most general statement of results. \par The authors make clear in the preface that they consider that the strength of a theory is tested by its usefulness in dealing with important examples and this belief influences the presentation of the material. The book is divided into four parts: Part I, Basic Theory; Part II, Dimension Theory; Part III, Extensions; Part IV, Examples. \par Part I starts with a Chapter of examples: various ring theoretic constructions that preserve the noetherian condition are introduced and important examples such as Weyl Algebras, Group Rings and Enveloping Algebras of finite dimensional Lie algebras are put into the noetherian framework. This Chapter sets the scene for much of the later material, the reasons for later abstract developments can usually be found in problems that arise in the study of the examples in Chapter 1. The next three Chapters include a presentation of the Goldie theory and a discussion of the prime and semiprime spectra of noetherian rings that include such important topics as reduced rank, the additivity principle, patch continuity and localization theory. The final Chapter in Part I discusses noncommutative analogues of Dedekind theory. \par Part II consists of three Chapters, each dedicated to a particular dimension: Chapter 6, Krull dimension; Chapter 7, Global dimension; Chapter 8, Gelfand-Kirillov dimension. The main properties are developed and related to the examples introduced in Chapter 1. \par Part III is mainly concerned with extensions, $R\subseteq S$, of rings. In Chapter 9 a noncommutative version of the Nullstellensatz is established for a wide variety of rings including Weyl algebras, polycyclic-by-finite group rings and enveloping algebras. Chapter 10 studies the relations between the prime spectra of R and S. Chapter 11 discusses stability and cancellation theory for modules. The simplified proofs, due to Coutinho, of Stafford's important noncommutative generalizations of theorems of Bass, Serre and Forster-Swan are presented. Chapter 12 continues along the path started in chapter 11 by studying K-theory for extensions of rings, culminating in a proof of Quillen's Theorem on the K-groups of filtered rings. The authors make this discussion more accessible by concentrating on the most important group $K\sb 0.$ \par The final part, Part IV discusses three areas of application of the foregoing theory: polynomial identity rings, enveloping algebras and rings of differential operators on algebraic varieties. Of course, special cases of these have been discussed throughout earlier parts of the book in the guise of matrix algebras, skew polynomial rings and Ore extensions. The authors make no attempt to be comprehensive, but rather they indicate the main appearances of the noetherian theory. In fact, the first two areas already have one or more books devoted to their study while the third area is a subject that is still developing rapidly. One application that is not covered is that of group rings of polycyclic-by- finite groups; the authors refer to the excellent book by Passman on this subject. However, I feel sure that if publication of this book had been delayed for another year or so the authors would not have been able to resist including a treatment of {\it J. A. Moody}'s beautiful solution of the Goldie Rank conjecture [Bull. Am. Math. Soc., New. Ser. 17, 113-116 (1987; reviewed below)] which uses many of the ideas presented in this book. \par In conclusion, the book is well written and one is easily encouraged to read it. It is an essential possession for any serious worker in this area. Any research student that masters the contents will be well placed to produce good work. For the beginner the necessary pre-knowledge is about that of a student who has attended a basic first year graduate course in Algebra, together with some introductory course in Ring theory.
[T.H.Lenagan]

MSC 2000:: *16P40 Associative Noetherian rings and modules
16-02 Research monographs (assoc. rings and algebras)
17B35 Universal enveloping algebras (Lie algebras)
16U10 Integral domains (noncommutative)
16Rxx Associative rings with polynomial identity
16U30 Divisibility, noncommutative UFDs
16P60 Chain conditions on annihilators and summands
16S20 Centralizing and normalizing extensions
16E10 Homological dimensions (assoc. rings and algebras)
16E20 K-theory of noncommutative rings
16S34 Group rings (assoc. rings)

Keywords: Noetherian Ring Theory; Goldie's Theorem; Weyl Algebras; Enveloping Algebras of finite dimensional Lie algebras; noetherian rings; reduced rank; additivity principle; patch continuity; localization; Krull dimension; Global dimension; Gelfand-Kirillov dimension; Nullstellensatz; polycyclic-by-finite group rings; cancellation theory; K-theory; extensions; polynomial identity rings; rings of differential operators; skew polynomial rings; Ore extensions; Goldie Rank conjecture

Citations: Zbl 0644.16014

Cited in: Zbl 1237.16001 Zbl 1135.16001 Zbl 1006.16023 Zbl 0980.16019 Zbl 0957.16004 Zbl 0958.16024 Zbl 0931.16012 Zbl 0943.13020 Zbl 0890.16013 Zbl 0879.16011 Zbl 0872.16017 Zbl 0859.16020 Zbl 0762.17009 Zbl 0724.16011 Zbl 0719.16014 Zbl 0695.16012 Zbl 0691.16019 Zbl 0679.16001 Zbl 0693.16004

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]