Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring and $M$ is an $R$-module. There is an non-negative integer $t$ such that $M/(0 :_{M} \mathfrak{m}^t)$ is finitely generated. Then $$\dim M = \dim M/(0 :_{M} \mathfrak{m}^t)$$
I try to prove this proposition but i have not find out the proof. Can someone give me any idea to prove this? Thanks.