I have read that for a given finitely generated module $M$ over a ring $R$ and a multiplicative set $S$ in $R$, $\mathrm{Ann}(S^{-1}M)=S^{-1}\mathrm{Ann}(M)$ as ideals in $S^{-1}R. $ Is it also true for an infinitely generated module? If not, what is a counterexample? Here, $\mathrm{Ann}(M)=\{r\in R:rm=0 \ \forall \ m\in M\}$ and similarly $\mathrm{Ann}(S^{-1}M)$ is defined.
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Take $R = \mathbf{Z}$, $M = \mathbf{Q}/\mathbf{Z}$, and $S = \mathbf{Z}^\times$. Then $\mathop{\mathrm{Ann}}(M) = 0$, but $S^{-1}M = 0$, so $\mathop{\mathrm{Ann}}(S^{-1}M) = S^{-1}R = \mathbf{Q} \neq 0 = S^{-1}\mathop{\mathrm{Ann}}(M)$.
answered Sep 26, 2016 at 18:35