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.
PVD, atomic domain, F + M construction, root-closed, factor ring, condition ∗, antimatter domain