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If $R$ is a local ring with maximal ideal $m$ and the intersection of powers of $m$ is $0$, then the $m$-adic topology is metrizable. Is there a condition on $R$ assuring that the metric space so formed is complete? It would also be interesting to know if there is such a condition which is not only sufficient but also necessary.

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  • $\begingroup$ What is the metric on the $m$-adic topology? $\endgroup$ – Manos Dec 5 '13 at 3:10
  • $\begingroup$ $R$ as a metric space? I'm not an expert at this, but it seems like this is equivalent to $R$ being $M$-adically complete, that is, the canonical homomorphism from $R$ into $\hat{R}$ is surjective. (It's already injective by your condition that the intersection of powers is $\{0\}$.) If you know this and are actually just looking for conditions to make it surjective, then I apologize and look forward to answers. $\endgroup$ – rschwieb Dec 5 '13 at 15:56
  • $\begingroup$ As you say this is the question. $\endgroup$ – Rodney Coleman Dec 6 '13 at 13:32

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