Considering A as a unique factorization domain, we must show that every prime ideal of A is generated by a set of prime elements. I was able to do it for a principal prime ideal, but I couldn't do it for other cases.
2026-03-25 23:24:45.1774481085
Show that every prime ideal of A (UFD) is generated by a set of prime elements
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Consider an element f of the prime ideal and factor it as a product of prime elements in the UFD A. Since the ideal is prime, one factor must actually belong to the ideal and collecting these elements as f runs through the ideal gives you a generating set as desired.
Note that the case of a principal ideal is the very definition of being prime: an element is a prime element if and only if the principal ideal generated by it is prime.