Modules over valuation domains.

*(English)*Zbl 0578.13004
Lecture Notes in Pure and Applied Mathematics, Vol. 97. New York-Basel: Marcel Dekker, Inc. XI, 317 p. (1985).

This is a half-brother of the first author’s treatise: ”Infinite abelian groups”, Vol. I (1970; Zbl 0209.055) and Vol. II (1973; Zbl 0257.20035)]. In addition to modules over a valuation domain (VD), modules over a commutative domain or a valuation, i.e. uniserial, ring (VR) are also discussed. Many of the results, some new, are due to the authors (and their coworkers). E. Matlis, R. B. Warfield (et al.) are also well represented. The book is intended for both experts and graduate students, and assumes acquaintance with commutative rings (with topology), modules, homological algebra.

Chapters I, II, III develop basics (e.g. equivalence of purity and relative divisibility for submodules over a Prüfer domain). Chapter IV relates projective dimension to cardinality of generating sets and properties of chains.

Chapter V: Topology. Homological aspects of completeness, Matlis duality. Ultracompleteness of a filtration F (cosets of submodules in F with finite intersection property do intersect) characterizes injectives. - Chapter VI: Divisibility, injectivity. - Chapter VII: Standard uniserials. Existence of non-standard uniserials and of a VR not image of a VD. End(U) for U uniserial. Direct sum of uniserials. - Chapter VIII introduces heights and indicators. - Chapter IX relates decomposition of finite generated and polyserial modules to number of generators, Goldie dimension, pure composition series. - Chapter \(X: \alpha\)-invariants and basic submodules. - Chapter XI: Decomposition of a pure-injective (involves summand with trivial \(\alpha\)-invariants). - Chapter XII: Torsion-completeness. Cotorsion modules \((Ext^ 1(F,C)=0\) for all torsion-free F). - Chapter XIII: Torsion, separable modules. - Chapter XIV: Torsion-free modules: finite rank modules over an almost maximal VD, chains of pure submodules, slender modules.

There is much more. Also, a fair range of exercises and 26 research problems. The lively style of this book will surely inspire its readers.

Chapters I, II, III develop basics (e.g. equivalence of purity and relative divisibility for submodules over a Prüfer domain). Chapter IV relates projective dimension to cardinality of generating sets and properties of chains.

Chapter V: Topology. Homological aspects of completeness, Matlis duality. Ultracompleteness of a filtration F (cosets of submodules in F with finite intersection property do intersect) characterizes injectives. - Chapter VI: Divisibility, injectivity. - Chapter VII: Standard uniserials. Existence of non-standard uniserials and of a VR not image of a VD. End(U) for U uniserial. Direct sum of uniserials. - Chapter VIII introduces heights and indicators. - Chapter IX relates decomposition of finite generated and polyserial modules to number of generators, Goldie dimension, pure composition series. - Chapter \(X: \alpha\)-invariants and basic submodules. - Chapter XI: Decomposition of a pure-injective (involves summand with trivial \(\alpha\)-invariants). - Chapter XII: Torsion-completeness. Cotorsion modules \((Ext^ 1(F,C)=0\) for all torsion-free F). - Chapter XIII: Torsion, separable modules. - Chapter XIV: Torsion-free modules: finite rank modules over an almost maximal VD, chains of pure submodules, slender modules.

There is much more. Also, a fair range of exercises and 26 research problems. The lively style of this book will surely inspire its readers.

Reviewer: C.P.L.Rhodes

##### MSC:

13A18 | Valuations and their generalizations for commutative rings |

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

13F05 | Dedekind, Prüfer, Krull and Mori rings and their generalizations |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

13A05 | Divisibility and factorizations in commutative rings |

13J10 | Complete rings, completion |

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