There's a common result that says for a commutative ring $A$ with a multiplicatively closed subset $S$ and a given finitely generated $A$-module $M$ we have $S^{-1}M=0 \iff sM=0$ for some $s \in S$.
The proof is fairly straightforward but it relies on the existence of finitely many generators. (Simply by definition of localisation we obtain annhilators for each generator).
My question is as follows. What is an example for non finitely generated module $M$ wherein a zero localisation doesn't imply existence of annihilators and vice versa.
It feels as though the choice of $M$ should be pathological as any $\mathbb Z$ module having zero localisation would imply existence of $0\in S$ which would annhilate.
Would appreciate a hint here thanks.
Consider $\Bbb Q/\Bbb Z$ over $\Bbb Z$ with $S=\Bbb Z\setminus \{0\}$.
The existence of an annihilating element always implies zero localization.