Definition. Let $R$ be a Noetherian ring, $I$ a proper ideal, and $M$ a finite $R$-module. An ideal $J\subset I$ is called a reduction ideal of $I$ with respect to $M$ if $JI^nM = I^{n+1}M$ for some (or equivalently all) sufficiently large $n$.
Question. Let $(R,m)$ be a Noetherian local ring and $M$ be a finite faithful $R$-module. let $I$ be an ideal of $R$ such that $IM=mM$. can we say that $I$ is a reduction ideal of $m$?($I$ is a reduction ideal of $m$ if $Im^n=m^{n+1}$ for some sufficiently large $n$)
if not what conditions can we add (on $I$ or $R$ or...) to can say $I$ is a reduction ideal of $m$?
thank you.