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I have read a couple proofs that that the intersection of all prime ideals contains only nilpotent elements that use a claim like this:

Suppose that $a$ is an element of $A$ that is not nilpotent. Let $S$ be the set of ideals of $A$ that do not contain any element of the form $a^n$. Since $(0) \in S$, $S$ is not empty; then by Zorn's Lemma, $S$ has a maximal element $\mathfrak{m}$.

However, it's not clear to me that every chain of ideals in $S$ has an upper bound, which appears necessary to use Zorn's Lemma. Am I missing something obvious?

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The union of elements of such a chain is an ideal and an upper bound.

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