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## Krull’s intersection theorem for commutative local not necessarily noetherian rings

Is there a characterisation of those commutative local not necessarily noetherian rings that satisfy Krull's intersection theorem ? How can the intersection theorem be phrased in terms the module category of the ring or its completion ?

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 Wouldn't it just mean that $R\to \hat{R}$ is injective? – Donu Arapura Sep 9 2011 at 16:17 Or the given ring $R$ is Hausdorff in the $M$-adic topology (where $M$ is its maximal ideal). – Jesse Elliott Sep 9 2011 at 18:10 Someone should retag this, but I don't want to bump it to the top and it's not clear to me which tags are appropriate. – David White Sep 9 2011 at 20:08