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I've been trying without success to figure out what are the rings $R$ such that whenever $M_n, n \in \omega$ is a countably infinite collection of pairwise distinct maximal ideals then $\bigcap_{n \in \omega}M_n=0$. If $R$ is a Dedekind domain then this obviously holds, and if $R$ has this property and has infinitely many maximal ideals then it has to have zero radical. Thanks for any input or hint.

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    What if all ideals are equal ?2012-12-02
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    Maximal ideals $M_n$ are supposed to be distinct, I should have probably written pairwise distinct.2012-12-02
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    A related notion is a semiprimitive ring, which is a ring such that the Jacobson radical is zero.2012-12-02
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    The set $\{0,1,2,\cdots\}$.2012-12-02

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