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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.

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Completion with respect to...? I know you know what you are thinking, but it's best not to leave these things unsaid :) –  rschwieb May 27 '13 at 22:21

2 Answers 2

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.

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Thanks. I found it in Theorem 8.12. –  Paul Kuhn May 28 '13 at 5:06

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!)

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