Ideal systems. An introduction to multiplicative ideal theory.

*(English)*Zbl 0953.13001
Pure and Applied Mathematics, Marcel Dekker. 211. New York, NY: Marcel Dekker. xii, 422 p. (1998).

Many results in commutative ring theory depend only on the multiplicative structure of the ring. The author’s philosophy is that such results should be derived as far as possible without making reference to the additive structure of the ring. This point of view goes back to P. Lorenzen in 1939 and is probably best known to commutative ring theorists through P. Jaffard’s book “Les systèmes d’idéaux” (1960; Zbl 0101.27502), which is written in the language of ordered abelian groups.

The book under review gives a modern treatment of multiplicative ideal theory, valid for both commutative rings and monoids, in the language of ideal systems on commutative monoids. The first part of the book is based on the concept of \(x\)-ideals of K. E. Aubert and is valid for commutative rings, topological and graded commutative rings, and commutative monoids. The second part deals with multiplicative ideal theory on cancellative commutative monoids which corresponds to the integral domain case.

Some of the topics covered in the book include the monoid versions of Krull’s intersection theorem and principal ideal theorem, valuation rings, Dedekind domains, Prüfer domains, PVMDs, Krull domains, and factorization theory in integral domains. In some cases the results are explicitly stated for both monoids and integral domains, while in other cases the translation to rings is left to the reader. However, some standard topics in commutative algebra which depend on the additive structure of the ring, such as dimension theory, polynomial and power series rings, and homological aspects of commutative ring theory, are not covered.

The book has 27 chapters, each of which has historical notes with a guide to the literature and exercises with solutions. This book is a welcome addition to the literature and gives a very readable, updated development of multiplicative ideal theory in the context of commutative monoids. It should be of interest for those in commutative ring theory, number theory, and semigroup theory.

The book under review gives a modern treatment of multiplicative ideal theory, valid for both commutative rings and monoids, in the language of ideal systems on commutative monoids. The first part of the book is based on the concept of \(x\)-ideals of K. E. Aubert and is valid for commutative rings, topological and graded commutative rings, and commutative monoids. The second part deals with multiplicative ideal theory on cancellative commutative monoids which corresponds to the integral domain case.

Some of the topics covered in the book include the monoid versions of Krull’s intersection theorem and principal ideal theorem, valuation rings, Dedekind domains, Prüfer domains, PVMDs, Krull domains, and factorization theory in integral domains. In some cases the results are explicitly stated for both monoids and integral domains, while in other cases the translation to rings is left to the reader. However, some standard topics in commutative algebra which depend on the additive structure of the ring, such as dimension theory, polynomial and power series rings, and homological aspects of commutative ring theory, are not covered.

The book has 27 chapters, each of which has historical notes with a guide to the literature and exercises with solutions. This book is a welcome addition to the literature and gives a very readable, updated development of multiplicative ideal theory in the context of commutative monoids. It should be of interest for those in commutative ring theory, number theory, and semigroup theory.

Reviewer: David F.Anderson (Knoxville)

##### MSC:

13A15 | Ideals and multiplicative ideal theory in commutative rings |

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

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

20M12 | Ideal theory for semigroups |

20M14 | Commutative semigroups |

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

13F30 | Valuation rings |

13G05 | Integral domains |