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For what commutative rings with infinitely many maximal ideals we can say that the intersection of any combination of finitely many maximal ideals is not zero? Obviously it holds for Dedekind domains because a product of fin many is not zero and the product is contained in the intersection. I would like some weaker condition... Thanks

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Equally obviously, you can drop the word "Dedekind".... –  Steven Landsburg Mar 7 at 13:52

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If finitely many distinct maximal ideals $\mathfrak{m}_1, ..., \mathfrak{m}_r$ of a commutative ring $R$ have intersection $0$, then by the Chinese remainder theorem the ring is isomorphic to the finite product of fields $\prod_{i=1}^r R/\mathfrak{m}_i$; in particular, it has only finitely many maximal ideals. So the answer to your question is: "For all."

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Every Noetherian ring which is not Artinian; In other words every Noetherian ring of positive dimension. (because THEOREM 7.30 of R.Y.Sharp book "Steps in Commutative Algebra, Second edition" says :
Let $G$ be a module over the ring $R$, and assume that $G$ is annihilated by the product of finitely many (not necessarily distinct) maximal ideals of $R$, Then $G$ is a Noetherian $R-module$ if and only if $G$ is an Artinian $R-module$

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