Pseudo-almost valuation domains are quasilocal going-down domains, but not conversely

Abstract

Ayman Badawi has recently introduced the PAVDs, a class of (commutative integral) domains which is found strictly between the class of APVDs (“almost pseudo valuation domains”) and that of the (necessarily quasilocal) domains having a linearly ordered prime spectrum. It is known that the latter class strictly contains the class of quasilocal going-down domains; it is proved that the class of quasilocal going-down domains strictly contains the class of PAVDs. Consequently, each seminormal PAVD is a divided domain. Moreover, for each n, 1 ≤ n ≤ ∞, an example is constructed of a divided domain (necessarily a quasilocal going-down domain) of Krull dimension n which is not a PAVD. Keywords Pseudo-almost valuation domain, Prime ideal, Going-down domain, Divided domain, Quasilocal, Valuation overring, Root extension, Seminormal, D+M construction, Krull dimension Mathematics Subject Classification (2000) Primary 13B24, 13G05, Secondary 13A15, 13F05