Let $R$ be a commutative ring with $1$, and let $M$ be a faithful multiplication module (namely, $xM=0$ for $x\in R$ gives $x=0$, and any submodule equals $IM$ for some ideal of $R$). If $IM=I^2M$, is it true that $I=I^2$ necessarily?
Of course, the faithfulness is a necessary condition as seen in this answer. Thanks for any suggestion!