∗-Almost super-homogeneous ideals in ∗-h-local domains

Abstract

In this paper, we introduce ∗-almost independent rings of Krull type (∗-almost IRKTs) and ∗-almost generalized Krull domains (∗-almost GKDs) in the general theory of almost factoriality, neither of which need be integrally closed. This fills a gap left in [D. D. Anderson and M. Zafrullah, On∗-Semi-Homogeneous Integral Domains, Advances in Commutative Algebra (Springer, Singapore, 2019)]. We characterize them by ∗-almost super-SH domains, where a domain [Formula: see text] is called a ∗-almost super-SH domain if every nonzero proper principal ideal of [Formula: see text] is a ∗-product of ∗-almost super-homogeneous ideals. We prove that (1) a domain [Formula: see text] is a ∗-almost IRKT if and only if [Formula: see text] is a ∗-almost super-SH domain, (2) a domain is a ∗-almost GKD if and only if [Formula: see text] is a type 1 ∗-almost super-SH domain and (3) a domain [Formula: see text] is a ∗-almost IRKT and an AGCD-domain if and only if [Formula: see text] is a ∗-afg-SH domain. Further, we characterize them by their integral closures. For example, we prove that a domain [Formula: see text] is an almost IRKT if and only if [Formula: see text] is a root extension with [Formula: see text] [Formula: see text]-linked under [Formula: see text] and [Formula: see text] is an IRKT. Examples are given to illustrate the new concepts.