Factoring Ideals in Almost Dedekind Domains and Generalized Dedekind Domains

Abstract

We start with an overview of the rings for which every proper ideal is a product of radical ideals, rings introduced by Vaughan and Yeagy under the name of SP-rings. The integral domains with this property are called here domains with radical factorization. We give several characterizations of this type of integral domains by revisiting, completing and generalizing the work by Vaughan–Yeagy (Canad. J. Math. 30:1313–1318, 1978) and Olberding (Arithmetical properties of commutative rings and monoids, Chapman & Hall/CRC, Boca Raton, 2005). In Sect. 3.2, we study almost Dedekind domains having the property that each nonzero finitely generated ideal can be factored as a finite product of powers of ideals of a factoring family (definition given below). In the subsequent section, we provide a review of the Prufer domains in which the divisorial ideals can be factored as a product of an invertible ideal and pairwise comaximal prime ideals, after papers by Fontana–Popescu (J. Algebra 173:44–66, 1995), Gabelli (Commutative Ring Theory, Marcel Dekker, New York, 1997) and Gabelli–Popescu (J. Pure Appl. Algebra 135:237–251, 1999). The final section is devoted to the presentation of various general constructions due to Loper–Lucas (Comm. Algebra 31:45–59, 2003) for building examples of almost Dedekind (non Dedekind) domains of various kinds (e.g., almost Dedekind domains which do not have radical factorization or which have a factoring family for finitely generated ideals or which have arbitrary sharp or dull degrees (definitions given below)).