Let $M$ be a faithful finite $A$-module, for a (commutative unitary) ring $A$; assume also that there are two ideals, $I\subseteq J\subset A$, such that $IM=JM$. Show that $I=J$.

Let $\{m\_1,\dots ,m\_n\}\subseteq M$ be a set of generators of $M$. Then for every $a\in J\setminus I$, exist $a\_{i1},\dots ,a\_{in}\in I$ such that $am\_i=a\_{i1}m\_1+\dots +a\_{in}m\_n$, for all $1\le i\le n$. So I have a $n\times n$ matrix defined by $(a\_{ij})$. What I should prove is, likely, that there is a non-zero $b\in J$ such that $b\in \operatorname {Ann} M$. How do I recover it though?