This is an exercise from hungerford p.140 and I am totally stuck at it. I denote $S$ to be the collection of all units and all products of primes in the given integral domain $R$. And I showed that $S$ is closed under multiplication and division. Now I assume there is an element in $R-S$. Then the ideal generated by the element is disjoint from $S$. Now I think of a collection of ideals disjoint from $S$. Then by Zorn's lemma there is a maximal element in the collection. Now I want to show that the maximal element is a prime ideal. But I cannot proceed from this point at all and am totally stuck.... Could anyone help me? Or is there some completely different solution?
2026-04-07 16:31:57.1775579517
An integral domain whose every nonzero prime ideal contains a nonzero principal ideal that is prime is UFD.
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