Prove that a $UFD$ is a $PID$ if and only if every prime ideal is maximal.
I know a proof but used Zorn's lemma.Is there a proof that doesn't require Zorn's lemma?
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Duplicate: If in a UFD every maximal ideal is principal then it is a PID (see this answer) – YuiTo Cheng Mar 16 '20 at 14:34
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Does this answer your question? Prove that a UFD is a PID if and only if every nonzero prime ideal is maximal – Sam Mar 16 '20 at 14:37