Q:

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

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.

A:

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 $M=0$.

