Modules over local ring and completion I'm stuck again at a commutative algebra question. Would love some help with this completion business...
We have a local ring $R$ and $M$ is a $R$-module with unique assassin/associated prime the maximal ideal $m$ of $R$. 
i) prove that $M$ is also naturally a module over the $m$-adic completion $\hat{R}$, and $M$ has the same $R$ and $\hat{R}$-submodules.
ii) if $M$ and $N$ are two modules as above, show that $Hom_{R}(M,N) = Hom_{\hat{R}}(M,N)$. 
Best wishes and a happy new year!

Proposition. Let $R$ be a noetherian ring and $M$ an $R$-module, $M\neq 0$. Then $\operatorname{Ass}(M)=\{\mathfrak m\}$ iff for every $x\in M$ there exists a positive integer $k$ such that $\mathfrak m^kx=0$.
Proof. "$\Rightarrow$" Let $x\in M$, $x\neq 0$. Then $\operatorname{Ann}(x)$ is an ideal of $R$. Let $\mathfrak p$ be a minimal prime ideal over $\operatorname{Ann}(x)$. Then $\mathfrak p\in\operatorname{Ass}(M)$, so $\mathfrak p=\mathfrak m$. This shows that $\operatorname{Ann}(x)$ is an $\mathfrak m$-primary ideal, hence there exists a positive integer $k$ such that $\mathfrak m^k\subseteq\operatorname{Ann}(x)$.
"$\Leftarrow$" Let $\mathfrak p\in\operatorname{Ass}(M)$. The there is $x\in M$, $x\neq 0$, such that $\mathfrak p=\operatorname{Ann}(x)$. On the other side, on knows that there exists $k\ge 1$ such that $\mathfrak m^kx=0$. It follows that $\mathfrak m^k\subseteq \mathfrak p$, hence $\mathfrak p=\mathfrak m$.
Now take $x\in M$ and $a\in\hat R$. One knows that $\mathfrak m^kx=0$ for some $k\ge 1$. Since $R/\mathfrak m^k\simeq \hat R/\hat{\mathfrak m^k}$ and $\hat{\mathfrak m^k}=\mathfrak m^k\hat R$, there exists $\alpha\in R$  such that $a-\alpha\in \mathfrak m^k\hat R$. (Maybe Ted's answer is more illuminating at this point.) Now define $ax=\alpha x$. (This is well defined since $\mathfrak m^kx=0$.) Now both properties are clear.
