Finitely generated module whose completion has finite length

Question

Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m).$

If $n\ge 1$ is an integer such that $\widehat {\mathfrak m^n}\widehat M=0,$ then is it true that $\mathfrak m^n M=0$ ? Here $\widehat {(-)}$ denotes $\mathfrak m$-adic completion. In other words, I'm asking that if the completion of a finitely generated $R$-module has finite length over the completion of the ring, then did the module have finite length over $R$ to begin with?

Please help.

Answer 1

Notice that $R \to R^{\wedge}$ is faithfully flat. So, $R \otimes_R N \cong N \to R^\wedge \otimes_R N \cong N^\wedge$ is injective for any finitely generated $R$-module $N$. Now let $N = m^n M$. Since $N^\wedge$ is zero, $N$ is zero.