Let $R$ be a commutative ring, $S\subset R$ be a multiplicative closed subset. Let $M$ be a R-module (may not be f.g.).
By simple deduction, we have:
$S^{-1}M=0 \Leftrightarrow \forall m\in M,\ \mathrm{Ann}_R(m)\cap S\neq \emptyset$.
The problem is, do we have the following statement
$S^{-1}M=0\Leftrightarrow \mathrm{Ann}_R(M)\cap S\neq\emptyset$ ?
When $M$ is finitely generated, we do have it and it's proved in this question: