Finitely generated modules over complete ring is zero if

Question

Let $M$ be a finitely generated module over $\hat{R}$ which is $R$ completed with respect to a finitely generated ideal $I$. If $M \otimes \frac{R}{I^{n}} = 0$ for every $n$, is it true that $M = 0$? If not, is it true when $R$ is noetherian?

Answer 1

Hint: Since $M\otimes R/I^n\simeq M/I^nM$ then $M=I^nM$ for every $n$. Now use that $M$ is an inverse limit of the system ${M/I^nM}$.