##
**Noncommutative Noetherian rings. With the cooperation of L. W. Small.**
*(English)*
Zbl 0644.16008

Pure and Applied Mathematics. A Wiley-Interscience Publication. Chichester etc.: John Wiley & Sons. XV, 596 p.; £65.00 (1987).

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.

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.

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.

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.

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 \(K_ 0.\)

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 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.

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.

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.

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.

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.

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 \(K_ 0.\)

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 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.

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.

Reviewer: T.H.Lenagan

### MSC:

16P40 | Noetherian rings and modules (associative rings and algebras) |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

17B35 | Universal enveloping (super)algebras |

16U10 | Integral domains (associative rings and algebras) |

16Rxx | Rings with polynomial identity |

16U30 | Divisibility, noncommutative UFDs |

16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |

16S20 | Centralizing and normalizing extensions |

16E10 | Homological dimension in associative algebras |

16E20 | Grothendieck groups, \(K\)-theory, etc. |

16S34 | Group rings |