Bulletin of the
Korean Mathematical Society BKMS

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 $x, y\in R_{\Nil(R)}$ and $(xy)\phi(P)\subseteq \phi(P)$, then there exists an integer $m\geqslant 1$ such that either $x^m\in \phi(R)$ or $y^m\phi(P)\subseteq \phi(P)$. 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.

Keywords: $\phi$-ring, valuation domain, pseudo-valuation ring, almost valuation ring, $\phi$-PAVR, chained ring

MSC numbers: 13A15, 13A10, 13F05