Star-Operations Induced by Overrings

Abstract

Let $D$ be an integral domain with quotient field $K$. A star-operation $\star$ on $D$ is a closure operation $A \longmapsto A^\star$ on the set of nonzero fractional ideals, $F(D)$, of $D$ satisfying the properties: $(xD)^\star = xD$ and $(xA)^\star = xA^\star$ for all $x \in K^\ast$ and $A \in F(D)$. Let ${\M S}$ be a multiplicatively closed set of ideals of $D$. For $A \in F(D)$ define $A_{\M S} = \{x \in K \mid xI \subseteq{A}$, for some $I \in {\M S}\}$. Then $D_{\M S}$ is an overring of $D$ and $A_{\M S}$ is a fractional ideal of $D_{\M S}$. Let ${\M S}$ be a multiplicative set of finitely generated nonzero ideals of $D$ and $A \in F(D)$, then the map $A \longmapsto A_{\M S}$ is a finite character star-operation if and only if for each $I \in {\M S}$, $I_v = D$. We give an example to show that this result is not true if the ideals are not assumed to be finitely generated. In general, the map $A \longmapsto A_{\M S}$ is a star-operation if and only if $\bar {\M S}$, the saturation of ${\M S}$, is a localizing GV-system. We also discuss star-operations given of the form $A \longmapsto \cap AD_\alpha$, where $D = \cap D_\alpha$.