# 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.

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 $$M=0$$.