Commutative semigroup rings. (English) Zbl 0566.20050

Gilmer, Robert

Chicago Lectures in Mathematics. Chicago-London: The University of Chicago Press. XII, 380 p. £10.25; $ 13.75 (1984).

The book gives the first systematic presentation of the research done in the theory of commutative semigroup algebras. Chapter 1 presents some basic commutative semigroup theory needed in the sequel. Some general techniques and special elements (units, nilpotents, idempotents, zero- divisors) of semigroup rings are discussed in Chapter 2. Two subsequent chapters deal with more global properties of monoid rings (e.g. integral closure, chain conditions). They aim to find necessary and sufficient conditions on the coefficient ring and the monoid in order that the monoid ring should satisfy a given property. The last chapter discusses the Krull dimension and the uniqueness of coefficients problem for monoid rings. A good presentation and bibliographical comments closing each section make the book very useful not only for commutative algebra and semigroup theorists but also for those studying related non-commutative structures.

Reviewer: J.Okniński

MSC:

20M25	Semigroup rings, multiplicative semigroups of rings
20M14	Commutative semigroups
20-02	Research exposition (monographs, survey articles) pertaining to group theory
13-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
13C15	Dimension theory, depth, related commutative rings (catenary, etc.)
13Exx	Chain conditions, finiteness conditions in commutative ring theory

Keywords:

commutative semigroup algebras; units; nilpotents; idempotents; semigroup rings; monoid rings; chain conditions; Krull dimension; uniqueness of coefficients problem