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> Let $R$ be a commutative ring with identity such that $R$ has exactly one prime ideal $P$.
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> Prove: all elements in $P$ are nilpotent.
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While doing this problem, I used the fact that "the nilradical of $R$ is equal to the intersection of all prime ideals of $R$" (in this case, the intersection of all prime ideals $=P$), then I can solve this problem.

However, it seems to me that this fact overkills this problem because we have a strong condition that there is only one prime ideal.

I am here to ask if there is a much more simple and direct approach (which I have overlooked) to solving this problem.

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Let me put it in another way: is there any proof without using Zorn's lemma?