Pseudo-Dedekind and Strong Pseudo-Dedekind Factorization

Abstract

The present chapter is devoted to the study of integral domains having two other kinds of ideal factorization. An integral domain is said to have strong pseudo-Dedekind factorization if each proper ideal can be factored as the product of an invertible ideal (possibly equal to the ring) and a finite product of pairwise comaximal prime ideals with at least one prime in the product. On the other hand, an integral domain is said to have pseudo-Dedekind factorization if each nonzero noninvertible ideal can be factored as the product of an invertible ideal (which might be equal to the ring) and finitely many pairwise comaximal primes. We observe that an integral domain with pseudo-Dedekind factorization has strong factorization (Sect. 4.1) and an integrally closed domain with pseudo-Dedekind factorization is an h-local Prufer domain. Nonintegrally closed local domains with pseudo-Dedekind factorization are fully described in terms of pullbacks of valuation domains. Several characterizations of integral domains with strong pseudo-Dedekind factorization are also given. In particular, we show that an integral domain has strong pseudo-Dedekind factorization if and only if it is an h-local generalized Dedekind domain. Finally, we investigate the ascent and descent of several types of ideal factorizations from an integral domain R to the Nagata ring R(X) and vice versa.