Give an arbitrary commutative ring (not necessarily noetherian) $A$ with $\mathfrak{m}$ a maximal ideal. Is the completion $\hat{A}$ w.r.t. the ideal $\mathfrak{m}$ a local ring? If so, is the maximal ideal of $\hat{A}$ generated by $\mathfrak{m}$, i.e., is it equal to $\mathfrak{m}\hat{A}$?

I know that this is true when $A$ is noetherian.