On the converse of a well-known fact about Krull domains

Kaplansky-Type Theorems in Graded Integral Domains

“…It is shown in [18] that a domain D is a UFD (resp, π-domain, Krull domain) if and only if every t-ideal of D is a t-product of principal (resp., invertible, t-invertible) prime ideals. The following two results are the graded domain analogues of these results.…”

Almost factoriality of integral domains and Krull-like domains

“…It is known that D is a Krull domain if and only if each nonzero prime ideal of D contains a t-invertible prime ideal [Kang 1989a, Theorem 3.6]. Also, we know that D is a Krull domain if and only if D is a Mori PvMD [Kang 1989a, Theorem 3.2].…”

“…After Zafrullah's paper [1985], several types of almost divisibility of integral domains have been studied, for example, AB-domains, AP-domains, APvMDs, API-domains and ADdomains (see Section 1). Recall from [Kang 1989a] that D is a Krull domain (resp., UFD) if and only if for every nonzero ideal I of D, I t is t-invertible (resp., principal). In this paper, we define D to be 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).…”

Integral Domains With a Free Semigroup of *-Invertible Integral *-Ideals

“…Thus P = P i for some i, and so P is a * -invertible prime * -ideal. □ Let X 1 (R) be the set of height-one prime ideals of R. An integral domain R is a Krull domain if (i) R P is a rank-one DVR for each P ∈ X 1 (R), (ii) R = ∩ P ∈X 1 (R) R P , and (iii) each nonzero element of R is contained in finitely many height-one prime ideals of R. It is well known that R is a Krull domain if and only if each nonzero proper principal ideal of R is a t-product of (tinvertible) prime ideals [17,Theorem 3.9].…”

“…Thus Q is principal because R M is quasi-local. We remark that if we take * = d in Theorem 3.7, then it is well known that the statements in Theorem 3.7 are equivalent to R being a π-domain ( [1,17]). In the case of * = t (resp., w), it follows from [17, Theorem 3.9] (resp., [18,Theorem 3.6]) that the equivalent conditions in Theorem 3.7 are equivalent to R being a Krull domain.…”