Almost factoriality of integral domains and krull-like domains

Let D be an integral domain, D̄ be the integral closure of D, and γ be a numerical semigroup with γ ⊆ N₀. 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 ({α_α}) of D, there exists a positive integer n = n{α_α} such that ({α_αⁿ})_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 ⊆ D̄ is a bounded root extension, then D is an AK-domain (resp., AUF-domain) if and only if D̄ is a Krull domain (resp., Krull domain with torsion t-class group) and D is t-linked under D̄. We also prove that if D is a Krull domain (resp., UFD) with char.D ≠ 0, then the (numerical) semigroup ring D[γ] is a nonintegrally closed AK-domain (resp., AUF-domain).