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**Noncommutative Noetherian rings. With the cooperation of L. W. Small.
Reprinted with corrections from the 1987 original.**
*(English)*
Zbl 0980.16019

Graduate Studies in Mathematics. 30. Providence, RI: American Mathematical Society (AMS). xx, 636 p. (2001).

The first edition of this book [Pure and Applied Mathematics, Wiley-Interscience Publication, Chichester etc. (1987; Zbl 0644.16008)] now has become a standard reference on Noetherian rings. According to the authors’ preface, the changes are minor, also there are some extra references to more recent developments. The book is well written and perfectly suits for both specialists and students beginning to study ring theory. An essentially cheaper edition was highly awaited by the latter. The exposition allows to read the chapters of this book relatively independent.

Apart from Chapter 0 summarizing some prerequisites, the book consists of four big parts, these parts are divided into 15 Chapters. Part I is called Basic theory. It starts with important examples and constructions of rings, such as Weyl algebras, enveloping algebras, group rings, skew polynomial rings, Laurent polynomials, etc. The authors demonstrate how certain examples of Noetherian rings arise in particular contexts, descriptions of other basic properties of that rings are given. The first major step in a structure theory for noncommutative Noetherian rings is Goldie’s theorem on the structure of rings of fractions for prime and semiprime rings. The next three Chapters systematically expose Goldie’s theorem, its consequences and extensions such as prime and semiprime spectra of Noetherian rings, Morita contexts, Artin-Rees property, and localization theory. Finally, Chapter 5 discusses noncommutative analogues of Dedekind rings and hereditary Noetherian rings.

Part II is called Dimensions and consists of three chapters. Each chapter is devoted to a particular dimension. To any Noetherian ring or module one can attach the Krull dimension, this is a measure of how close the ring or module is to being Artinian. The next is the global dimension that measures the deviation of a ring from being semisimple. Here the authors expose basic facts without going far into homological algebra. The Gelfand-Kirillov dimension is a measure of the rate of growth of an algebra in terms of an arbitrary generating set, this dimension measures the deviation from finite dimensionality. These dimensions are scrutinized for the forgoing examples and constructions.

Part III is called Extensions and is mainly concerned with extensions of rings \(R\subset S\). Chapter 9 studies a noncommutative analogue of the Nullstellensatz. Chapter 10 describes prime ideals of certain extension rings \(S\) in terms of those of the original ring \(R\). Chapter 11 is concerned with stability and cancellation theory for modules. It includes Stafford’s important noncommutative generalizations of theorems of Bass, Serre and Forster-Swan. Chapter 12 studies \(K\)-theory for extensions of rings.

The final Part IV is called Examples. It discusses three big areas of applications of the foregoing theory. The authors do not try to make a comprehensive account but just illustrate applications of the theory of Noetherian rings. Chapter 13 is about the structure and properties of polynomial identity (PI) rings. The exposition includes central polynomials, Azumaya algebras, trace rings, Kurosh conjecture and catenary property for prime ideals. Chapter 14 is about universal enveloping algebras of finite dimensional Lie algebras, mainly it concentrates on the theory in solvable and completely solvable cases. The final Chapter 15 is about rings of differential operators on algebraic varieties, the approach provides an elementary introduction to this area.

Apart from Chapter 0 summarizing some prerequisites, the book consists of four big parts, these parts are divided into 15 Chapters. Part I is called Basic theory. It starts with important examples and constructions of rings, such as Weyl algebras, enveloping algebras, group rings, skew polynomial rings, Laurent polynomials, etc. The authors demonstrate how certain examples of Noetherian rings arise in particular contexts, descriptions of other basic properties of that rings are given. The first major step in a structure theory for noncommutative Noetherian rings is Goldie’s theorem on the structure of rings of fractions for prime and semiprime rings. The next three Chapters systematically expose Goldie’s theorem, its consequences and extensions such as prime and semiprime spectra of Noetherian rings, Morita contexts, Artin-Rees property, and localization theory. Finally, Chapter 5 discusses noncommutative analogues of Dedekind rings and hereditary Noetherian rings.

Part II is called Dimensions and consists of three chapters. Each chapter is devoted to a particular dimension. To any Noetherian ring or module one can attach the Krull dimension, this is a measure of how close the ring or module is to being Artinian. The next is the global dimension that measures the deviation of a ring from being semisimple. Here the authors expose basic facts without going far into homological algebra. The Gelfand-Kirillov dimension is a measure of the rate of growth of an algebra in terms of an arbitrary generating set, this dimension measures the deviation from finite dimensionality. These dimensions are scrutinized for the forgoing examples and constructions.

Part III is called Extensions and is mainly concerned with extensions of rings \(R\subset S\). Chapter 9 studies a noncommutative analogue of the Nullstellensatz. Chapter 10 describes prime ideals of certain extension rings \(S\) in terms of those of the original ring \(R\). Chapter 11 is concerned with stability and cancellation theory for modules. It includes Stafford’s important noncommutative generalizations of theorems of Bass, Serre and Forster-Swan. Chapter 12 studies \(K\)-theory for extensions of rings.

The final Part IV is called Examples. It discusses three big areas of applications of the foregoing theory. The authors do not try to make a comprehensive account but just illustrate applications of the theory of Noetherian rings. Chapter 13 is about the structure and properties of polynomial identity (PI) rings. The exposition includes central polynomials, Azumaya algebras, trace rings, Kurosh conjecture and catenary property for prime ideals. Chapter 14 is about universal enveloping algebras of finite dimensional Lie algebras, mainly it concentrates on the theory in solvable and completely solvable cases. The final Chapter 15 is about rings of differential operators on algebraic varieties, the approach provides an elementary introduction to this area.

Reviewer: Victor Petrogradsky (Ulyanovsk)

### 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 |

16U20 | Ore rings, multiplicative sets, Ore localization |

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 |

16S30 | Universal enveloping algebras of Lie algebras |

16S32 | Rings of differential operators (associative algebraic aspects) |

16S35 | Twisted and skew group rings, crossed products |

16S36 | Ordinary and skew polynomial rings and semigroup rings |

16E60 | Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc. |

16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |

16N60 | Prime and semiprime associative rings |

16W70 | Filtered associative rings; filtrational and graded techniques |

16P20 | Artinian rings and modules (associative rings and algebras) |

16P50 | Localization and associative Noetherian rings |

16P90 | Growth rate, Gelfand-Kirillov dimension |