Exercise 9, § 2

Let $A$ be a Noetherian ring, $m$ a maximal ideal of $A$ and $M$ an $A$-module such that $\{0\}$ is $m$-primary in $M$. If $\hat{A}$ is the Hausdorff completion of $A$ with respect to the $m$-adic topology, show that the canonical mapping $M \to M \otimes_A \hat{A}$ is bijective (consider first the case where $M$ is finitely generated, using Proposition 7 of no. 5; in the general case, consider $M$ as the direct limit of its finitely generated submodules).