ON ϕ-PSEUDO ALMOST VALUATION RINGS

Abstract

The purpose of this paper is to introduce a new class of rings that is closely related to the classes of pseudo valuation rings (PVRs) and pseudo-almost valuation domains (PAVDs). A commutative ring R is said to be <TEX>${\phi}$</TEX>-ring if its nilradical Nil(R) is both prime and comparable with each principal ideal. The name is derived from the natural map <TEX>${\phi}$</TEX> from the total quotient ring T(R) to R localized at Nil(R). A prime ideal P of a <TEX>${\phi}$</TEX>-ring R is said to be a <TEX>${\phi}$</TEX>-pseudo-strongly prime ideal if, whenever <TEX>$x,y{\in}R_{Nil(R)}$</TEX> and <TEX>$(xy){\phi}(P){\subseteq}{\phi}(P)$</TEX>, then there exists an integer <TEX>$m{\geqslant}1$</TEX> such that either <TEX>$x^m{\in}{\phi}(R)$</TEX> or <TEX>$y^m{\phi}(P){\subseteq}{\phi}(P)$</TEX>. If each prime ideal of R is a <TEX>${\phi}$</TEX>-pseudo strongly prime ideal, then we say that R is a <TEX>${\phi}$</TEX>-pseudo-almost valuation ring (<TEX>${\phi}$</TEX>-PAVR). Among the properties of <TEX>${\phi}$</TEX>-PAVRs, we show that a quasilocal <TEX>${\phi}$</TEX>-ring R with regular maximal ideal M is a <TEX>${\phi}$</TEX>-PAVR if and only if V = (M : M) is a <TEX>${\phi}$</TEX>-almost chained ring with maximal ideal <TEX>$\sqrt{MV}$</TEX>. We also investigate the overrings of a <TEX>${\phi}$</TEX>-PAVR.