On the saturations of submodules

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 $A_{[\tau] }= \left\{x\in S\colon A \text{ for some } t\in \tau\right\}$, 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 $AS= BS = S$.