Let $R$ be a commutative graded ring, $m$ be its graded maximal ideal, $M$ be a finitely generated graded module over $R$.
A homogeneous ideal $I\subseteq m$ is a $M$-reduction of $m$ if $Im^{n-1}M=m^{n}M$ for some $n\in \mathbb{N}$.
Then people claimed that a homogeneous ideal $I\subseteq m$ is a $M$-reduction of $m$ if and only if $(M/IM)_{n}=0$ for $n\gg 0$.
I could not prove it. Please help me. Thanks.