Module of dimension zero over a local ring. > Let $(A,m)$ be a Noetherian local ring and let $M$ be an $A$-module of dimension $0$ ($\dim M=\dim A/\operatorname{Ann}M$). Then every element of $M$ is annihilated by a power of $m$. I would appreciate some help proving this. Thanks.

Since $\dim M=0$ we have $\dim A/\operatorname{Ann}M=0$, so $A/\operatorname{Ann}M$ is a local artinian ring and you are done.