Almost factoriality of integral domains and Krull-like domains

Abstract

Let D be an integral domain, D be the integral closure of D, and be a numerical semigroup with ‫ގ‬ 0 . Let t be the so-called t-operation on D. We will say that D is an AK-domain (resp., AUF-domain) if for each nonzero ideal ({a α }) of D, there exists a positive integer n = n({a α }) such that ({a n α }) t is t-invertible (resp., principal). In this paper, we study several properties of AK-domains and AUF-domains. Among other things, we show that if D ⊆ D is a bounded root extension, then D is an AK-domain (resp., AUFdomain) if and only if D is a Krull domain (resp., Krull domain with torsion t-class group) and D is t-linked under D. We also prove that if D is a Krull domain (resp., UFD) with char( D) = 0, then the (numerical) semigroup ring D[ ] is a nonintegrally closed AK-domain (resp., AUF-domain).