Ideal factorization in strongly discrete independent rings of Krull type

Abstract

Let [Formula: see text] be an integral domain and [Formula: see text] be the so-called [Formula: see text]-operation on [Formula: see text]. In this paper, we define the notion of [Formula: see text]-ZPUI domains which is a natural generalization of ZPUI domains introduced by Olberding in 2000. We say that [Formula: see text] is a [Formula: see text]-ZPUI domain if every nonzero proper [Formula: see text]-ideal [Formula: see text] of [Formula: see text] can be written as [Formula: see text] for some [Formula: see text]-invertible ideal [Formula: see text] of [Formula: see text] and [Formula: see text] is a nonempty collection of pairwise [Formula: see text]-comaximal prime [Formula: see text]-ideals of [Formula: see text]. Then, among other things, we show that [Formula: see text] is a [Formula: see text]-ZPUI domain if and only if the polynomial ring [Formula: see text] is a [Formula: see text]-ZPUI domain, if and only if [Formula: see text] is a strongly discrete independent ring of Krull type. We construct three types of new [Formula: see text]-ZPUI domains from an old one by [Formula: see text]-construction, pullback, and [Formula: see text]-domains. We also show that given an abelian group [Formula: see text], there is a ZPUI domain with ideal class group [Formula: see text] but not a Dedekind domain. Finally, we introduce and study the notion of [Formula: see text]-ISP domains as a generalization of [Formula: see text]-ZPUI domains.