TITLE: Artinian rings and associated prime ideals
QUESTION [5 upvotes]: Let $R$ be a commutative ring with unity. Show that $R$ is artinian ring if and only if there exists a finite length $R$-module $M$, such that
  $$\{r\in R \mid rm=0 ,\forall m\in M\}=(0).$$

The only thing I can think of is that I need to find $M$ such that $\mathrm{Ass}_{R}M=(0)$, but I'm not sure how to do this. $\mathrm{Ass}_{R}M$ is the set of all associated prime ideals of $M$.

REPLY [3 votes]: Well, one direction is easy.  If $R$ is artinian then take $M = R$.
For the other direction , the given $M$ is both artinian and noetherian because it has finite length.  As it's noetherian take a finite set of generators $m_i$ and define $R \to M^n$ by $a \mapsto (am_i)$.  The condition on the annihilator of $M$ makes this injective so $R$ is a submodule of an artinian module and hence artinian.