NOTE ON PSEUDO-VALUATION DOMAINS WHICH ARE NOT VALUATION DOMAINS

Abstract

In this article, we discuss the n-root closedness, root closedness, seminormality, S-root closedness, S-closedness, F-closedess of PVDs. A valuation domain, being integrally closed, is obviously root closed. So our interest of study is for a class of non-valuation PVDs. Let R ⊂ B be a domain extension such that R is a PVD and the common ideal P of R and B is a prime ideal in R. If R is n-root closed (respectively root closed, seminormal, S-root closed, S-closed, F-closed) in B, then R/P is PVD, which is n-root closed (respectively root closed, seminormal, S-root closed, S-closed, F-closed) in B/P. Further we study the relationship of atomic PVDs to atomic PVDs, SHFDs, LHFDs and BVDs. We also discuss a relative ascent and descent in general and particularly for the antimatter property of PVDs.