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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.

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As far as I know graded rings don't have only one maximal graded ideal unless they are *local. Moreover, I'm not so sure that what you say really works if $I$ can't be generated by homogeneous elements of degree $1$. –  user26857 Dec 8 '12 at 16:24

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