Annihilators and exact sequence

Question

Let $R$ be a commutative ring and $0\longrightarrow L \overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}N\longrightarrow 0$ be an exact sequence of $R$-modules. How to prove $$\mbox{Ann}(L)\mbox{Ann}(N) \subset \mbox{Ann}(M).$$

And I know that the following does not hold in general $$\mbox{Ann}(L)\mbox{Ann}(N) \supseteq \mbox{Ann}(M).$$ It is sufficient to consider the following example

Let $R=k[x,y]$, and $L=(x,y)/(y),\quad M=k[x,y]/(y),\quad N=k[x,y]/(x,y).$

Answer 1

$aL=0$ and $bN=0$ implies $(ab)x=0$ for all $x\in M$:

$bg(x)=0\Rightarrow g(bx)=0\Rightarrow bx\in\ker g\Rightarrow bx\in\operatorname{im}f\Rightarrow bx=f(y)\Rightarrow abx=af(y)=f(ay)=0$.