Let $R$ be a commutative ring. Let $M_1$, $M_2$ and $M_3$ be $R$-modules. Let the following sequence be exact:
$$0\longrightarrow M_1 \overset{f}{\longrightarrow}M_2\overset{g}{\longrightarrow}M_3\longrightarrow 0.$$
I have proven that
$$\mbox{Ann}(M_1)\mbox{Ann}(M_3) \subset \mbox{Ann}(M_2).$$
And I know that the following does not hold in general
$$\mbox{Ann}(M_2) \subset\mbox{Ann}(M_1)\mbox{Ann}(M_3).$$
I am looking for a counterexample.