The Stacks project

Exercise 111.7.1. Let $A$ be a ring. Let $S \subset A$ be a multiplicative subset. Let $M$ be an $A$-module. Let $N \subset S^{-1}M$ be an $S^{-1}A$-submodule. Show that there exists an $A$-submodule $N' \subset M$ such that $N = S^{-1}N'$. (This useful result applies in particular to ideals of $S^{-1}A$.)