**Graduate Studies in Mathematics**

Volume: 30;
1987;
636 pp;
Hardcover

MSC: Primary 16;

Print ISBN: 978-0-8218-2169-5

Product Code: GSM/30

List Price: $89.00

Individual Member Price: $71.20

**Electronic ISBN: 978-1-4704-2084-0
Product Code: GSM/30.E**

List Price: $89.00

Individual Member Price: $71.20

# Noncommutative Noetherian Rings

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*J. C. McConnell; J. C. Robson*

This is an updated 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.

#### Table of Contents

# Table of Contents

## Noncommutative Noetherian Rings

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface to the Revised Edition xiii14 free
- Preface xv16 free
- Notation xix20 free
- Chapter 0. Preliminaries 122 free
- Part I. Basic Theory 930
- Chapter 1. Some Noetherian Rings 1132
- §1. Matrices 1132
- §2. Skew Polynomial Rings 1536
- §3. Weyl Algebras 1940
- §4. Skew Power Series and Laurent Polynomials 2142
- §5. Group Rings and Generalizations 2243
- §6. Skew Polynomial Rings in Several Variables 2647
- §7. Enveloping Algebras and Their Generalizations 3051
- §8. Further Simple Rings 3556
- §9. Additional Remarks 3859

- Chapter 2. Quotient Rings and Goldie's Theorem 4162
- Chapter 3. Structure of Semiprime Goldie Rings 6384
- Chapter 4. Semiprime Ideals in Noetherian Rings 97118
- Chapter 5. Some Dedekind-like Rings 141162

- Part II. Dimensions 183204
- Chapter 6. Krull Dimension 185206
- §1. Deviation of a Poset 186207
- §2. Krull Dimension of Modules 192213
- §3. Krull Dimension in Rings 200221
- §4. Prime Ideals and FBN Rings 204225
- §5. Bounds on Krull Dimensions 209230
- §6. Calculation of Krull Dimension 213234
- §7. Stable Bounds on Generators 219240
- §8. Quotient Rings and Localization 223244
- §9. Skew Polynomials over Commutative Rings 233254
- §10. Additional Remarks 242263

- Chapter 7. Global Dimension 245266
- §1. Preliminaries 245266
- §2. Change of Rings 250271
- §3. Factor Rings 253274
- §4. Localization 260281
- §5. Estimates of Global Dimension 261282
- §6. Filtered and Graded Modules 270291
- §7. Regular Rings 275296
- §8. Fixed Rings 278299
- §9. Skew Polynomial Rings 282303
- §10. Skew Polynomials over Commutative Rings 290311
- §11. Simple Dedekind Domains 291312
- §12. Additional Remarks 294315

- Chapter 8. Gelfand-Kirillov Dimension 297318

- Part III. Extensions 339360
- Part IV. Examples 475496
- Chapter 13. Polynomial Identity Rings 477498
- §1. Polynomial Identities 478499
- §2. Nilpotence 484505
- §3. Central Simple Algebras 486507
- §4. Embeddings and Matrix Rings 489510
- §5. Central Polynomials 492513
- §6. Semiprime Rings and Central Polynomials 497518
- §7. Prime Ideals and Azumaya Algebras 501522
- §8. Integral Extensions and Prime Rings 507528
- §9. The Trace Ring and Maximal Orders 514535
- §10. Affine k-algebras 518539
- §11. Additional Remarks 523544

- Chapter 14. Enveloping Algebras of Lie Algebras 525546
- §1. Basics 526547
- §2. Prime Ideals 530551
- §3. Eigenvalues and Prime Ideals 534555
- §4. Primitive Ideals 537558
- §5. The Solvable Case 542563
- §6. When Eigenvectors are Central 545566
- §7. When g is Algebraic 550571
- §8. The Simple Algebras A(V,δ,Γ) 552573
- §9. The General Case 557578
- §10. Additional Remarks 563584

- Chapter 15. Rings of Differential Operators on Algebraic Varieties 565586

- References 595616
- Index of Notation 621642 free
- Index 625646
- Back Cover Back Cover1658

#### Readership

Graduate students and research mathematicians interested in ring theory and its applications.

#### Reviews

A comprehensive list of references (containing more than 500 items) is included.

-- Mathematica Bohemica

A model of mathematical writing, as perfectly written a mathematics book as I have seen … It can be profitably read by non-experts … an almost perfectly conceived account of major developments and general methods … will remain a basic reference for many years …

-- Bulletin of the AMS

Very thorough and illuminating … A veritable tour de force, encompassing a wide range of topics in some depth … very easy to find information in this book … full of illuminating examples which throw a light on [the theory].

-- Proceedings of the Edinburgh Mathematical Society

Self-contained, comprehensive … The creation of this valuable resource is a service to mathematics …

-- Mathematical Reviews

An intrinsically interesting branch of algebra … Until … 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 … authors set out to fill this gap and have succeeded admirably … easy to read and use … well written … An essential possession for any serious worker in the area.

-- Zentralblatt MATH

An account of noncommutative Noetherian rings, giving the theory as far as it exists but with constant emphasis on constructions and examples. [This is] a daunting task but the authors have succeeded well … highly readable … well indexed … will rapidly become the standard text in the field and will stimulate further progress.

-- Bulletin of the LMS

An abundance of well-organized material … a must for those in the area.

-- International Mathematical News