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**Serial rings.**
*(English)*
Zbl 1032.16001

Dordrecht: Kluwer Academic Publishers. ix, 226 p. (2001).

This book presents an extensive and up-to-date overview of the structure theory of serial rings and their modules, especially pure projective modules and pure injective modules. An interesting feature of this book is that it includes several results on serial rings obtained by the Kiev school that previously seemed little known to experts. For instance, the book contains an elementary proof through a diagonalization of matrices of the Drozd-Warfield theorem stating that finitely presented modules over serial rings are direct sums of uniserial modules. The author gives also a short proof of the Kirichenko-Warfield theorem on the structure of two-sided Noetherian serial rings, and Kirichenko’s description of right Noetherian serial rings.

The book contains 17 chapters and a bibliography of 135 papers. The first nine chapters are devoted to the classical theory of serial rings, including decompositions of finitely presented modules over serial and uniserial rings, prime ideals and localizations in serial rings, structure of serial rings satisfying certain chain conditions – Artinian, Noetherian, ACC on annihilators, Krull dimension. In the remaining eight chapters, the author describes the structure of modules over serial rings, including model theory for modules, indecomposable pure injective modules over serial rings, super-decomposable pure injective modules over commutative valuation rings, pure injective modules over commutative valuation domains, pure projective modules over certain classes of uniserial rings, and \(\Sigma\)-pure injective modules over serial rings. Of particular interest is the construction of a pure projective module without an indecomposable decomposition over an exceptional uniserial ring.

This book contains a good number of exercises and interesting open problems, with commentaries on historical background of the material presented. The book will be a very useful source of reference for researchers in Ring and Module Theory, and can also be used as a textbook for graduate students.

The book contains 17 chapters and a bibliography of 135 papers. The first nine chapters are devoted to the classical theory of serial rings, including decompositions of finitely presented modules over serial and uniserial rings, prime ideals and localizations in serial rings, structure of serial rings satisfying certain chain conditions – Artinian, Noetherian, ACC on annihilators, Krull dimension. In the remaining eight chapters, the author describes the structure of modules over serial rings, including model theory for modules, indecomposable pure injective modules over serial rings, super-decomposable pure injective modules over commutative valuation rings, pure injective modules over commutative valuation domains, pure projective modules over certain classes of uniserial rings, and \(\Sigma\)-pure injective modules over serial rings. Of particular interest is the construction of a pure projective module without an indecomposable decomposition over an exceptional uniserial ring.

This book contains a good number of exercises and interesting open problems, with commentaries on historical background of the material presented. The book will be a very useful source of reference for researchers in Ring and Module Theory, and can also be used as a textbook for graduate students.

Reviewer: Nguyen Viet Dung (Zanesville)

### MSC:

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

16L30 | Noncommutative local and semilocal rings, perfect rings |

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

16D10 | General module theory in associative algebras |

16D40 | Free, projective, and flat modules and ideals in associative algebras |

16D50 | Injective modules, self-injective associative rings |

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

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

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