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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.

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closed as off-topic by YCor, Chris Godsil, Steven Sam, Stefan Kohl, S. Carnahan Jul 4 '14 at 23:22

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  • $\begingroup$ Yes. You can find those things in Bourbaki's Algèbre Commutative. $\endgroup$ – Vinteuil Jul 3 '14 at 12:19
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Yes for the first question : see Bourbaki Commutative Algebra III, § 2, Proposition 19. No for the second question : see loc. cit., exercise 12.

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