##
**Modules over non-Noetherian domains.**
*(English)*
Zbl 0973.13001

Mathematical Surveys and Monographs. 84. Providence, RI: American Mathematical Society (AMS). xvi, 613 p. (2001).

The authors of this impressive text are two of the leading researchers in the theory of modules over non-Noetherian domains. Fifteen years ago they had an earlier text [“Modules over valuation domains”, Lect. Notes Pure Appl. Math. 97 (New York-Basel 1985; Zbl 0578.13004)], but since then new methods have been developed which improve their earlier exposition and extend the theory to a wider class of domains. The theory presented in both texts originates in the study of abelian groups and, more generally, modules over Dedekind domains. However, dropping the Noetherian property of the domain gives rise to greater, more interesting, intricacies, often requiring cardinality and set-theoretic tools.

While the text concentrates on modules over commutative domains (in particular, Krull, coherent, Prüfer, and divisorial domains), there is much in the presentation that can be of use to ring theorists not working specifically in this area. For example, finite presentations, projective dimension, injectivity, and pure-injectivity are all developed ab initio, giving the reader an informed introduction to these topics.

There is a wealth of information for the specialist. Many results appear here in book form for the first time. While the authors’ style is “reader-friendly”, the main text is closely-spaced and complemented with exercises and compact notes at the end of each chapter. (These notes not only give historical background but also provide references, not listed in the bibliography, to other results not covered in the main text.)

We end the review of this tour-de-force with a listing of its chapter titles: Commutative domains and their modules; Valuation domains; Prüfer domains; More non-Noetherian domains; Finitely generated modules; Projectivity and projective dimension; Divisible modules; Topology and filtration; Injective modules; Uniserial modules; Heights, invariants and basic submodules; Polyserial modules; RD- and pure-injectivity; Torsion modules; Torsion-free modules of finite rank; Infinite rank torsion-free modules.

While the text concentrates on modules over commutative domains (in particular, Krull, coherent, Prüfer, and divisorial domains), there is much in the presentation that can be of use to ring theorists not working specifically in this area. For example, finite presentations, projective dimension, injectivity, and pure-injectivity are all developed ab initio, giving the reader an informed introduction to these topics.

There is a wealth of information for the specialist. Many results appear here in book form for the first time. While the authors’ style is “reader-friendly”, the main text is closely-spaced and complemented with exercises and compact notes at the end of each chapter. (These notes not only give historical background but also provide references, not listed in the bibliography, to other results not covered in the main text.)

We end the review of this tour-de-force with a listing of its chapter titles: Commutative domains and their modules; Valuation domains; Prüfer domains; More non-Noetherian domains; Finitely generated modules; Projectivity and projective dimension; Divisible modules; Topology and filtration; Injective modules; Uniserial modules; Heights, invariants and basic submodules; Polyserial modules; RD- and pure-injectivity; Torsion modules; Torsion-free modules of finite rank; Infinite rank torsion-free modules.

Reviewer: John Clark (Dunedin)

### MSC:

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

13Cxx | Theory of modules and ideals in commutative rings |

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

13G05 | Integral domains |

13C05 | Structure, classification theorems for modules and ideals in commutative rings |

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

13A05 | Divisibility and factorizations in commutative rings |

13F30 | Valuation rings |