It is a consequence of a theorem known as Krull’s Hauptidealsatz that every non-unit element in a Noetherian domain is contained in a prime ideal of height 1. Assuming this, prove that a Noetherian domain R is a UFD if every prime ideal of height 1 in R is principal.
I want to use the following characterization of UFDs:
- ACCP holds
- every irreducible element is prime
The first point is obvious since R is noetherian, but I couldn't prove the second. I do not know whether the way I try is correct or not. Please help me, thank you.