Question about Module "localization"

Question

I'm here to ask if any of you know a counterexample for this statement:

If $S^{-1}M=0$, then there is $s$ such that $sM=0$.

I already proved that if $M$ is finitely generated this is true. But my question is what happens in general? Once the same proof doesn't fit for the general case, I think it isn't true. However, I couldnt came up with a counterexample.

Thanks in advance.

Answer 1

How about $R=\Bbb Z$ as the ring, $M=\Bbb Q/\Bbb Z$ as the module, and $S$ the set of non-zero elements of $\Bbb Z$?