On Pseudo-Almost Valuation Domains

Abstract

Let R be an integral domain with quotient field K and integral closure R ′. Anderson and Zafrullah called R an “almost valuation domain” if for every nonzero x ∈ K, there is a positive integer n such that either x n ∈ R or x −n ∈ R. In this article, we introduce a new closely related class of integral domains. We define a prime ideal P of R to be a “pseudo-strongly prime ideal” if, whenever x, y ∈ K and xyP ⊆ P, then there is a positive integer m ≥ 1 such that either x m ∈ R or y m P ⊆ P. If each prime ideal of R is a pseudo-strongly prime ideal, then R is called a “pseudo-almost valuation domain” (PAVD). We show that the class of valuation domains, the class of pseudo-valuation domains, the class of almost valuation domains, and the class of almost pseudo-valuation domains are properly contained in the class of pseudo-almost valuation domains; also we show that the class of pseudo-almost valuation domains is properly contained in the class of quasilocal domains with linearly ordered prime ideals. Among the properties of PAVDs, we show that an integral domain R is a PAVD if and only if for every nonzero x ∈ K, there is a positive integer n ≥ 1 such that either x n ∈ R or ax −n ∈ R for every nonunit a ∈ R. We show that pseudo-almost valuation domains are precisely the pullbacks of almost valuation domains, we characterize pseudo-almost valuation domains of the form D + M, and we use this characterization to construct PAVDs that are not almost valuation domains. We show that if R is a Noetherian PAVD, then R has Krull dimension at most one and R ′ is a valuation domain; we show that every overring of a PAVD R is a PAVD iff R ′ is a valuation domain and every integral overring of R is a PAVD.