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