TITLE: Is the completion of a commutative Noetherian local ring Noetherian?
QUESTION [9 upvotes]: Maybe for some straightforward, but not for me: 

Let $(R,\mathfrak{m})$ be  a commutative Noetherian local ring with maximal ideal $\mathfrak{m}$. Why is the completion $\widehat{R}$ of $R$ with respect to the maximal ideal $\mathfrak{m}$ again a Noetherian ring?

Thanks.

REPLY [7 votes]: I assume you are talking about the completion of a local Noetherian ring $A$ with respect to the topology induced by its maximal ideal $m$. Then $\hat{A}$ is again Noetherian local ring with maximal ideal $m \hat{A}$. Reference: Matsumura's Commutative Ring Theory p. 63.

REPLY [4 votes]: Yes; if $A$ is a Noetherian ring, and $\mathfrak a$ is an ideal of $A$, then the $\mathfrak a$-adic completion of $A$ is Noetherian.  This is Theorem 10.26 in Atiyah-MacDonald. (I would write the proof here for you, but it is rather involved; hopefully, this standard reference will suffice!)