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I've already proven that if $P$ is a prime ideal in a ring $R$ and $I_j$ are ideals of $R$ then $I_1I_2\cdots I_r\subseteq P$ implies $I_j\subseteq P$ for some $j$. I want to know if this is true in the infinite case, i.e. if $$\bigcap\limits_{i=1}^\infty I_i\subseteq P$$ then $P\supseteq I_i$ for some $i$. I haven't been able to prove that it's true after considerable effort, but I can't come up with a counterexample.

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Hint: The ideals $(3)$, $(5)$, $(7)$, $(9)$, $(11)$, ... in $\mathbb{Z}$ have intersection $\{0\}$.

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