The Stacks project

Exercise 111.7.4. Let $A$ be a ring. Let $M$ be a finite $A$-module. Let $S \subset A$ be a multiplicative set. Assume that $S^{-1}M = 0$. Show that there exists an $f \in S$ such that the principal localization $M_ f = \{ 1, f, f^2, \ldots \} ^{-1}M$ is zero.