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Suppose $R$ is a regular local ring and $I$ is a non-zero ideal such that $I$ is a radical ideal and $I$ is height unmixed. Suppose $J$ is any radical ideal contained in $I$ and with the same height as $I$. Can we always find a prime $P\in \operatorname{Ass}(R/J)$ such that $P$ is not minimal.

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$I=(x) \cap (y)$, $J = (x) \cap (y) \cap (z)$.

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Thanks, that was quite easy. – messi May 8 '13 at 7:35

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