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**An introduction to noncommutative Noetherian rings.
2nd ed.**
*(English)*
Zbl 1101.16001

London Mathematical Society Student Texts 61. Cambridge: Cambridge University Press (ISBN 0-521-83687-5/hbk; 0-521-54537-4/pbk; 0-511-20834-0/e-book). xxiv, 344 p. £ 22.99, $ 34.99/pbk; £ 50.00, $ 85.00/hbk; $ 32.00/e-book (2004).

The first edition of this book [An introduction to noncommutative Noetherian rings. Lond. Math. Soc. Student Texts 16. (Cambridge etc.: Cambridge University Press) (1989; Zbl 0679.16001)] now has become a standard textbook on Noetherian rings. After appearance of the first edition the explosive growth of the area of quantum groups provided a big array of examples of Noetherian rings and thus gave major impetus to research in Noetherian ring theory.

This book is rather an introduction to the subject in comparison with a more profound treatise which also recently appeared in the second edition [J. C. McConnell, J. C. Robson, Noncommutative Noetherian rings, Reprinted with corrections from the 1987 original, Graduate Studies in Mathematics 30, (Providence RI: AMS) (2001; Zbl 0980.16019)].

According to the preface, the book incorporates substantial revisions, especially in the first third of the book, where the presentation has been changed in favor of examples, constructions, and motivations. Opinions of students were taken into account in order to make reading easy for students that study the subject on their own. In particular, more examples with specific rings have been added to the early part of the book. A special attention has been paid to skew polynomial rings and skew Laurent rings. Their discussion starts with twists by automorphisms, then twist by derivations, and only then the general construction. It is shown that many interesting examples of enveloping algebras and quantum groups can be obtained by means of such extensions. Many examples are analyzed in details in the first two chapters (in both text and exercises) and are used repeatedly in later chapters to test new concepts and methods. Rings of fractions are obtained as rings of endomorphisms of appropriate modules thus avoiding tedious computations with equivalence classes of ordered pairs. This is done in accordance with the general philosophy of the book that noncommutative rings naturally arise as rings of operators. In place of systematic theory, quantum groups have been integrated into the book as examples to illustrate the theory. The book is supplied with an extensive selection of exercises.

The introductory chapter supplies notation, terminology, basic examples and constructions such as polynomial identity rings, group algebras, rings of differential operators, enveloping algebras, and examples of quantum groups. Apart from the introductory chapter, the book consists of 17 chapters.

The list of titles of the chapters is as follows. (1) A few Noetherian rings, (2) Skew polynomial rings, (3) Prime ideals, (4) Semisimple modules, Artinian modules, and torsionfree modules, (5) Injective hulls, (6) Semisimple rings of fractions, (7) Modules and semiprime Goldie rings, (8) Bimodules and affiliated prime ideals, (9) Fully bounded rings, (10) Rings and modules of fractions, (11) Artinian quotient rings, (12) Links between prime ideals, (13) The Artin-Rees property, (14) Rings satisfying the second layer condition, (15) Krull dimension, (16) Numbers of generators of modules, (17) Transcendental division algebras.

The exposition allows to read the chapters of this book relatively independently. Some bibliographical notes are included at the end of each chapter that supply the reader with comments on the origin of the results and further related references.

Finally, the appendix yields a list of open problems in the theory of Noetherian rings, the reader is supplied with information on (partial) solutions and developments on some of these problems since the appearance of the first edition.

The book is well written and perfectly suits for both students beginning to study Ring theory and established mathematicians that are just interested in learning basics of the theory of Noetherian rings.

This book is rather an introduction to the subject in comparison with a more profound treatise which also recently appeared in the second edition [J. C. McConnell, J. C. Robson, Noncommutative Noetherian rings, Reprinted with corrections from the 1987 original, Graduate Studies in Mathematics 30, (Providence RI: AMS) (2001; Zbl 0980.16019)].

According to the preface, the book incorporates substantial revisions, especially in the first third of the book, where the presentation has been changed in favor of examples, constructions, and motivations. Opinions of students were taken into account in order to make reading easy for students that study the subject on their own. In particular, more examples with specific rings have been added to the early part of the book. A special attention has been paid to skew polynomial rings and skew Laurent rings. Their discussion starts with twists by automorphisms, then twist by derivations, and only then the general construction. It is shown that many interesting examples of enveloping algebras and quantum groups can be obtained by means of such extensions. Many examples are analyzed in details in the first two chapters (in both text and exercises) and are used repeatedly in later chapters to test new concepts and methods. Rings of fractions are obtained as rings of endomorphisms of appropriate modules thus avoiding tedious computations with equivalence classes of ordered pairs. This is done in accordance with the general philosophy of the book that noncommutative rings naturally arise as rings of operators. In place of systematic theory, quantum groups have been integrated into the book as examples to illustrate the theory. The book is supplied with an extensive selection of exercises.

The introductory chapter supplies notation, terminology, basic examples and constructions such as polynomial identity rings, group algebras, rings of differential operators, enveloping algebras, and examples of quantum groups. Apart from the introductory chapter, the book consists of 17 chapters.

The list of titles of the chapters is as follows. (1) A few Noetherian rings, (2) Skew polynomial rings, (3) Prime ideals, (4) Semisimple modules, Artinian modules, and torsionfree modules, (5) Injective hulls, (6) Semisimple rings of fractions, (7) Modules and semiprime Goldie rings, (8) Bimodules and affiliated prime ideals, (9) Fully bounded rings, (10) Rings and modules of fractions, (11) Artinian quotient rings, (12) Links between prime ideals, (13) The Artin-Rees property, (14) Rings satisfying the second layer condition, (15) Krull dimension, (16) Numbers of generators of modules, (17) Transcendental division algebras.

The exposition allows to read the chapters of this book relatively independently. Some bibliographical notes are included at the end of each chapter that supply the reader with comments on the origin of the results and further related references.

Finally, the appendix yields a list of open problems in the theory of Noetherian rings, the reader is supplied with information on (partial) solutions and developments on some of these problems since the appearance of the first edition.

The book is well written and perfectly suits for both students beginning to study Ring theory and established mathematicians that are just interested in learning basics of the theory of Noetherian rings.

Reviewer: Victor Petrogradsky (Ulyanovsk)

### MSC:

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

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

16P50 | Localization and associative Noetherian rings |

16N60 | Prime and semiprime associative rings |

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

16D25 | Ideals in associative algebras |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

16S30 | Universal enveloping algebras of Lie algebras |

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

16S34 | Group rings |

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

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

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

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

16E10 | Homological dimension in associative algebras |

16N20 | Jacobson radical, quasimultiplication |

17B35 | Universal enveloping (super)algebras |