Pseudo-almost valuation domains are quasilocal going-down domains, but not conversely
This note proves a property of a pseudo-almost valuation domain (for short, PAVD), a concept defined and explored by \textit{A. Badawi} [Commun. Algebra 35, No. 4, 1167--1181 (2007; Zbl 1113.13001)] as follows. Let \(R\) be a domain with fraction field \(K\). For a prime ideal \(P\) of \(R\) one says that \(P\) is pseudo-strongly prime ideal if whenever \(xy P \subseteq P,\) with \(x,y \in K\), there exists \(n=n(x,y)\) such that \(x^n \in R\) or \(y^n P \subseteq P\). If each prime \(P\) of \(R\) is a pseudo-strongly prime ideal, one says that \(R\) is a pseudo-almost valuation domain (or PAVD). A going-down domain is a domain such that every overring of it satisfies the going-down property. The main result of the paper is that every PAVD is a quasilocal going-down domain. Furthermore, examples are provided to show that the class of quasilocal going-down domains sits strictly between the class of PAVDs and that of domains having linearly ordered primes.
scientific article; zbMATH DE number 5292721