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Let me paste the title:

Is there a nonzero module $M$ over an artinian local ring $(R,\mathfrak m)$ such that $\mathfrak mM=M\ ?$

Of course such a module could not be finitely generated.

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Such a nonzero module doesn't exist. If $(R,\mathfrak{m})$ is an Artinian local ring, then $\mathfrak{m}$ is nilpotent. If $\mathfrak{m}M=M$, then $M=\mathfrak{m}M=\mathfrak{m}^2M=\dots=\mathfrak{m}^nM=0$.

More generally, if $R$ is an Artinian ring, denote by $r$ the radical of $R$. Then $r$ is nilpotent. For any module $M$, if $rM=M$, then $M=0$.

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