Corollary

Let $M$ be an $A$-module and $a$ an ideal of $A$. The sub-$S^{-1}A$-modules $(S^{-1}a)(S^{-1}M)$, $a(S^{-1}M)$, $(S^{-1}a)j(M)$ (where $j : M \to S^{-1}M$ is the canonical mapping) and $S^{-1}(aM)$ of $S^{-1}M$ are identical. In particular, if $a$ and $b$ are two ideals of $A$, then