LOCALLY PSEUDO-VALUATION DOMAINS OF THE FORM D[X]_Nv

On n-absorbing ideals of locally divided commutative rings

“…( 3)). On the other hand, R is an LPVD and R ′ is a Dedekind domain by Corollary 24, so each overring of R(X) is an LPVD by [23,Corollary 3.9]. Therefore by Proposition 19 and Corollary 24, each overring of R(X) is a TAF-domain.…”

“…(1)⇒(3): Assume that R(X) is a TAF-domain. Then R is an LPVD, R ′ is a Prüfer domain and every overring of R(X) is an LPVD[23, Corollary 3.9]. Then R[X] Nv , being an overring of R(X), is a Mori LPVD by Corollary 24.…”

On Almost Pseudo-Valuation Domains, Ii

“…In [3], the authors introduced the notions of strongly primary ideals and almost PVDs as follows: an ideal I of D is strongly primary if, whenever xy ∈ I with x, y ∈ K implies x ∈ I or y n ∈ I for some integer n ≥ 1, while D is an almost PVD (APVD) if each prime ideal of D is strongly primary. They showed that if D is quasi-local with maximal ideal M , then D is an APVD if and only if there exists a valuation overring As in [9], we say that D is a locally pseudo valuation domain (LPVD) if D M is a PVD for all maximal ideals M of D. In [5], we studied when D[X] Nv is an LPVD. To do this, we introduced the notion of t-locally PVD; D is a t-locally PVD (t-LPVD) if D P is a PVD for all maximal t-ideals P of D. (Definitions related to the t-operation will be reviewed in the sequel.)…”