Star-Operations Induced by Overrings

Let $D$ be an integral domain with quotient field $K$. A star-operation $\star$ on $D$ is a closure operation $A⟼A^{\star }$ on the set of nonzero fractional ideals, $F(D)$, of $D$ satisfying the properties: $(xD{)}^{\star }=xD$ and $(xA{)}^{\star }=x{A}^{\star }$ for all $x\in K^{\ast }$ and $A \in F(D)$\mathrm{.}Let${\M S}$beamultiplicativelyclosedsetofidealsof$D$\mathrm{.}For$A \in F(D)$define$A_{\M S} = {x \in K \mid xI \subseteq{A}$,forsome$I \in {\M S}}$\mathrm{.}Then$D_{\M S}$isanoverringof$D$and$A_{\M S}$ is a fractional ideal of $D_{\text{M}S}$. Let $\text{M}S$ be a multiplicative set of finitely generated nonzero ideals of $D$ and $A\in F(D)$, then the map $A⟼A_{\text{M}S}$ is a finite character star-operation if and only if for each $I\in \text{M}S$, $I_v=D$. 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 $A \longmapsto A_{\M S}$isastar-operationifandonlyif$\bar {\M S}$, the saturation of $\text{M}S$, is a localizing GV-system. We also discuss star-operations given of the form $A⟼\cap A{D}_{\alpha }$, where $D = \cap D_\alpha$.