Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring and $M$ is an $R$module such that $\mathfrak{m}^tM=0$ for some nonnegative integer $t$. Then the length of $M$ is finite.
Is that right?
Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring and $M$ is an $R$module such that $\mathfrak{m}^tM=0$ for some nonnegative integer $t$. Then the length of $M$ is finite. Is that right? 

