ON ϕ-PSEUDO ALMOST VALUATION RINGS

Title & Authors
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 $\small{{\phi}}$-ring if its nilradical Nil(R) is both prime and comparable with each principal ideal. The name is derived from the natural map $\small{{\phi}}$ from the total quotient ring T(R) to R localized at Nil(R). A prime ideal P of a $\small{{\phi}}$-ring R is said to be a $\small{{\phi}}$-pseudo-strongly prime ideal if, whenever $\small{x,y{\in}R_{Nil(R)}}$ and $\small{(xy){\phi}(P){\subseteq}{\phi}(P)}$, then there exists an integer $\small{m{\geqslant}1}$ such that either $\small{x^m{\in}{\phi}(R)}$ or $\small{y^m{\phi}(P){\subseteq}{\phi}(P)}$. If each prime ideal of R is a $\small{{\phi}}$-pseudo strongly prime ideal, then we say that R is a $\small{{\phi}}$-pseudo-almost valuation ring ($\small{{\phi}}$-PAVR). Among the properties of $\small{{\phi}}$-PAVRs, we show that a quasilocal $\small{{\phi}}$-ring R with regular maximal ideal M is a $\small{{\phi}}$-PAVR if and only if V
Keywords
$\small{{\phi}}$-ring;valuation domain;pseudo-valuation ring;almost valuation ring;$\small{{\phi}}$-PAVR;chained ring;
Language
English
Cited by
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