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The Stacks project

Lemma 10.35.3. Let $R$ be a ring. If every prime ideal of $R$ is the intersection of the maximal ideals containing it, then $R$ is Jacobson.

Proof. This is immediately clear from the fact that every radical ideal $I \subset R$ is the intersection of the primes containing it. See Lemma 10.17.2. $\square$

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