Commutative rings with zero divisors. (English) Zbl 0637.13001

The idea for this monograph is to collect and collate results that have appeared in mathematical journals (mostly since 1970) on commutative rings containing zero divisors, and which can be proved by ideal- theoretic methods. It also contains some new material of the author and D. D. Anderson. That there is almost no overlap with other texts is due to the author’s quoting freely from earlier related texts. The general plan is that each of the six chapters is followed by a couple of pages of notes which give the historical setting for the material of the chapter, others’ alternative terminology and credit to the authors for the results of the chapter and of related results. The book ends with a handy reference to a list of 140 results contained in the text and an extensive list of references to books and papers.
The first chapter (Total quotient rings) introduces properties A and (a.c), gives characterizations of von Neumann regular rings and, the analogue, \(\pi\)-regular rings in the non-reduced case; the problem of the compactness of Min(R) is also discussed. - Chapter II (Valuation rings) introduces valuation/paravaluation rings via mappings onto/into ordered groups, and deals with Prüfer rings defined as rings for which every finitely generated regular ideal is invertible. Krull rings and the divisor class group are restricted to rings with the Marot property that every regular ideal is generated by its regular elements. These considerations lead naturally to integral closures in chapter III.


13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
13A18 Valuations and their generalizations for commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials