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If $(R,m)$ is a complete local ring (with respect to the $m-$adic topology) and $I$ a prime ideal in $R$, is $R/I$ complete (with respect to the $m/I-$adic topology)? It seems too strong, but I am unable to give a counterexample.

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    complete for which topology?2012-03-07
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    @seporhau: Clarified. I thought, it was convention to assume "with respect to the maximal ideal" unless otherwise specified.2012-03-07
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    What you'd like to be able to do is take a Cauchy sequence in $R/I$, then lift each element so that you have a Cauchy sequence in $R$, then take the limit and push it back down to the quotient. So the question becomes: under what conditions can you lift your Cauchy sequence mod $I$ to a Cauchy sequence in $R$?.2012-03-07
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    The completion functor from the category of $R$-modules to the catogory of $\hat R$-modules is exact. So the quotient by an ideal $I \subset R=\hat{R}$ is complete if and only if $I$ is complete as an $\hat R$-module. I tend to think that this is always the case at least for Noetherian $R$.2012-03-07
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    @DimaSustretov: Thank you.2012-03-08

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