##
**Flat covers of modules.**
*(English)*
Zbl 0860.16002

Lecture Notes in Mathematics. 1634. Berlin: Springer. x, 161 p. (1996).

In this monograph the author presents the properties of envelopes and covers for a given class of modules over an associative ring \(R\) with identity.

In Chapter 1 is given the minimal set of concepts, notations and results in the theory of modules and rings, used in the book. Also, it introduces the definitions of envelopes and covers and their elementary properties, it is proved that every module over an integral domain has a torsion free covering from which it follows that every module over a Prüfer domain admits a flat cover agreeing with its torsion free covering and in the last section are discussed the direct sums of envelopes and covers.

In Chapter 2, using the assumption that a certain class of modules is closed under direct limits, it is developed a general technique to solve the problem of existence, by manipulating generators of extension sequences. As an application it is proved the existence of injective envelopes, projective covers, pure injective envelopes and injective covers in a uniform way. It is shown that the existence of injective covers of left modules over the ring \(R\) is equivalent to \(R\) being left Noetherian and that the existence of nonzero injective covers of every nonzero module implies that \(R\) must be Artinian.

The main results in Chapter 3 are the existence of flat covers and cotorsion envelopes over a right coherent ring of finite weak global dimension. It is proved that over a right coherent ring \(R\), if a left \(R\)-module \(M\) has finite pure injective dimension, then \(M\) has a flat cover. Assuming the existence of flat covers, a relative homological theory is developed by using flat resolutions.

Chapter 4 deals with flat covers over commutative rings. Necessary notations and preliminary results for modules over commutative Noetherian rings are given, flat cotorsion modules using completions of certain free modules of flat modules are described, it is shown that all modules over commutative Noetherian rings of finite Krull dimension admit flat covers, the flat covers of Matlis reflexive modules are considered and it is shown that the existence of nonzero flat preenvelopes leads to a characterization of commutative Artinian rings.

In Chapter 5, as an application of the theory of flat covers developed in the previous chapters, the dual Bass numbers are defined by using minimal flat resolutions to describe modules over Gorenstein rings, strongly cotorsion and strongly torsion free modules are introduced and described by the vanishing property of the Bass numbers and the dual Bass numbers, also the Foxby classes and Gorenstein projective, injective and flat modules as well as their envelopes and covers are presented.

This monograph is suitable as a reference for researchers and graduate students who have interest in the general theory of covers and envelopes, in the theory of rings and modules and homological methods in commutative algebra.

In Chapter 1 is given the minimal set of concepts, notations and results in the theory of modules and rings, used in the book. Also, it introduces the definitions of envelopes and covers and their elementary properties, it is proved that every module over an integral domain has a torsion free covering from which it follows that every module over a Prüfer domain admits a flat cover agreeing with its torsion free covering and in the last section are discussed the direct sums of envelopes and covers.

In Chapter 2, using the assumption that a certain class of modules is closed under direct limits, it is developed a general technique to solve the problem of existence, by manipulating generators of extension sequences. As an application it is proved the existence of injective envelopes, projective covers, pure injective envelopes and injective covers in a uniform way. It is shown that the existence of injective covers of left modules over the ring \(R\) is equivalent to \(R\) being left Noetherian and that the existence of nonzero injective covers of every nonzero module implies that \(R\) must be Artinian.

The main results in Chapter 3 are the existence of flat covers and cotorsion envelopes over a right coherent ring of finite weak global dimension. It is proved that over a right coherent ring \(R\), if a left \(R\)-module \(M\) has finite pure injective dimension, then \(M\) has a flat cover. Assuming the existence of flat covers, a relative homological theory is developed by using flat resolutions.

Chapter 4 deals with flat covers over commutative rings. Necessary notations and preliminary results for modules over commutative Noetherian rings are given, flat cotorsion modules using completions of certain free modules of flat modules are described, it is shown that all modules over commutative Noetherian rings of finite Krull dimension admit flat covers, the flat covers of Matlis reflexive modules are considered and it is shown that the existence of nonzero flat preenvelopes leads to a characterization of commutative Artinian rings.

In Chapter 5, as an application of the theory of flat covers developed in the previous chapters, the dual Bass numbers are defined by using minimal flat resolutions to describe modules over Gorenstein rings, strongly cotorsion and strongly torsion free modules are introduced and described by the vanishing property of the Bass numbers and the dual Bass numbers, also the Foxby classes and Gorenstein projective, injective and flat modules as well as their envelopes and covers are presented.

This monograph is suitable as a reference for researchers and graduate students who have interest in the general theory of covers and envelopes, in the theory of rings and modules and homological methods in commutative algebra.

Reviewer: I.Crivei (Cluj-Napoca)

### MSC:

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

13-02 | Research exposition (monographs, survey articles) pertaining to commutative algebra |

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

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

13C10 | Projective and free modules and ideals in commutative rings |

13C11 | Injective and flat modules and ideals in commutative rings |

16E10 | Homological dimension in associative algebras |

13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |

18G05 | Projectives and injectives (category-theoretic aspects) |