On $phi$-pseudo almost valuation rings

摘要

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 a $phi$-ring if its nilradical $Nil(R)$ is both prime and comparable with each principal ideal. The name is derived from the natural map $phi$ from the total quotient ring $T(R)$ to $R$ localized at $Nil(R)$. A prime ideal $P$ of a $phi$-ring $R$ is said to be a {em $phi$-pseudo-strongly prime ideal} if, whenever $x, yin R_{Nil(R)}$ and $(xy)phi(P)subseteq phi(P)$, then there exists an integer $mgeqslant 1$ such that either $x^min phi(R)$ or $y^mphi(P)subseteq phi(P)$. If each prime ideal of $R$ is a $phi$-pseudo strongly prime ideal, then we say that $R$ is a {em $phi$-pseudo-almost valuation ring} ($phi$-PAVR). Among the properties of $phi$-PAVRs, we show that a quasilocal $phi$-ring $R$ with regular maximal ideal $M$ is a $phi$-PAVR if and only if $V=(M:M)$ is a $phi$-almost chained ring with maximal ideal $sqrt{MV}$. We also investigate the overrings of a $phi$-PAVR.