On the saturations of submodules

Abstract

Let \(R\subseteq S\) be a ring extension, and let \(A\) be an \(R\)-submodule of \(S\). The saturation of \(A\) (in \(S\)) by \(\tau\) is set \(A_{[\tau] }= \left\{x\in S : tx\in A  \text{ for some } t\in \tau\right\}\), where \(\tau\) is a multiplicative subset of \(R\). We study properties of saturations of \(R\)-submodules of \(S\). We use this notion of saturation to characterize star operations \(\star\) on ring extensions \(R\subseteq S\) satisfying the relation \((A\cap B)^{\star} = A^{\star}\cap B^{\star}\) whenever \(A\) and \(B\) are two \(R\)-submodules of \(S\) such that \(AS= BS = S\).