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**Torsion theories.**
*(English)*
Zbl 0657.16017

Pitman Monographs and Surveys in Pure and Applied Mathematics, 29. Harlow, Essex: Longman Scientific and Technical; New York: John Wiley & Sons, Inc. XVIII, 651 p.; $ 175.00 (1986).

This book is a comprehensive (651 pages) presentation of (hereditary) torsion theory and is designed to update the previous standard books in the area by the author [Localization of noncommutative rings (1975; Zbl 0302.16002)] and B. Stenström [Rings of quotients (1975; Zbl 0296.16001)]. It also has some overlap with the more specialized book of T. Albu and C. Năstăsescu [Relative finiteness in module theory (1984; Zbl 0556.16001)]. All major areas of torsion theory are discussed in detail with the exception of splitting torsion theories, decomposition functors, and some topics related to noncommutative algebraic geometry. The presentation assumes that the reader has a solid background in ring theory as well as some exposure to homological algebra.

After two introductory chapters in which the author presents the definition of a torsion theory via the “equivalence class of injectives” approach, the author sets out to study various classes of modules associated with torsion theories. These classes include the torsion modules, torsionfree modules, cotorsionfree modules, modules with relative chain conditions, cocritical and semicocritical modules, relatively projective and injective modules, relatively neat modules, and relatively injective modules. The standard types of submodules associated with a torsion theory are also considered, e.g., dense and relatively pure submodules.

The third portion of the book deals with functors naturally defined by torsion theories. The localization functor is the main example, but the relative Jacobson radical, the relative socle, and colocalization functors each have a chapter devoted to their study. The fourth part of the book has seven chapters devoted to the lattice structure of the set of torsion theories of a ring R. Much use is made of the fact that this lattice is a frame and thus has pseudocomplements. An investigation is made of the irreducible, prime, and semiprime elements of the lattice.

Next, special types of torsion theories are considered. These include semisimple torsion theories, symmetric torsion theories. TTF classes (Jansian torsion theories), centrally splitting torsion theories, torsion theories whose associated filters have finiteness conditions, torsion theories whose localization functor is exact, and perfect torsion theories. Part six of the book deals with change-of-rings properties of torsion theory. In particular, the transfer of torsion theories and their properties between two rings related by a ring homomorphism is discussed. Transfers by the use of Morita contexts are also discussed. Part seven gives several types of rings that can be characterized through the use of torsion theories. The stable rings (i.e., the rings for which every torsion class is closed under injective envelopes) is a major example. Other important classes are the left semi-Noetherian rings (the rings with Gabriel dimension) and the left semi-Artinian rings. Next some topics motivated by the classical study of commutative rings are considered. Torsion-theoretical concepts depending on the support and assassin of a module are studied. Torsion-theoretical primary decompositions of modules are studied. The relationship of prime torsion theories and prime ideals is examined. The final part of the book discusses relative local cohomology, derived functors for torsion theories, and the use of Cousin complexes.

Each chapter contains exercises that serve as a method of presenting specialized results. A very complete list of approximately 1000 papers on the subject provides an excellent guide to the literature. A section of historical notes also provides an easy source reference for all of the results in the book. The index provides for easy location of most technical terms. This book will probably become a standard general reference listing for persons working in the area of torsion theories.

After two introductory chapters in which the author presents the definition of a torsion theory via the “equivalence class of injectives” approach, the author sets out to study various classes of modules associated with torsion theories. These classes include the torsion modules, torsionfree modules, cotorsionfree modules, modules with relative chain conditions, cocritical and semicocritical modules, relatively projective and injective modules, relatively neat modules, and relatively injective modules. The standard types of submodules associated with a torsion theory are also considered, e.g., dense and relatively pure submodules.

The third portion of the book deals with functors naturally defined by torsion theories. The localization functor is the main example, but the relative Jacobson radical, the relative socle, and colocalization functors each have a chapter devoted to their study. The fourth part of the book has seven chapters devoted to the lattice structure of the set of torsion theories of a ring R. Much use is made of the fact that this lattice is a frame and thus has pseudocomplements. An investigation is made of the irreducible, prime, and semiprime elements of the lattice.

Next, special types of torsion theories are considered. These include semisimple torsion theories, symmetric torsion theories. TTF classes (Jansian torsion theories), centrally splitting torsion theories, torsion theories whose associated filters have finiteness conditions, torsion theories whose localization functor is exact, and perfect torsion theories. Part six of the book deals with change-of-rings properties of torsion theory. In particular, the transfer of torsion theories and their properties between two rings related by a ring homomorphism is discussed. Transfers by the use of Morita contexts are also discussed. Part seven gives several types of rings that can be characterized through the use of torsion theories. The stable rings (i.e., the rings for which every torsion class is closed under injective envelopes) is a major example. Other important classes are the left semi-Noetherian rings (the rings with Gabriel dimension) and the left semi-Artinian rings. Next some topics motivated by the classical study of commutative rings are considered. Torsion-theoretical concepts depending on the support and assassin of a module are studied. Torsion-theoretical primary decompositions of modules are studied. The relationship of prime torsion theories and prime ideals is examined. The final part of the book discusses relative local cohomology, derived functors for torsion theories, and the use of Cousin complexes.

Each chapter contains exercises that serve as a method of presenting specialized results. A very complete list of approximately 1000 papers on the subject provides an excellent guide to the literature. A section of historical notes also provides an easy source reference for all of the results in the book. The index provides for easy location of most technical terms. This book will probably become a standard general reference listing for persons working in the area of torsion theories.

### MSC:

16S90 | Torsion theories; radicals on module categories (associative algebraic aspects) |

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

16P50 | Localization and associative Noetherian rings |

16Nxx | Radicals and radical properties of associative rings |