# Noetherian domain R is a UFD if every prime ideal of height 1 in R is principal.

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:

1. ACCP holds
2. 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.

• Isn't this obvious? If $q$ is irreducible and contained in a principal prime $(p)$ then $q=pa$. Can you conclude from this? – user26857 Dec 3 '17 at 8:24
• @user26857 thanks.I got it. – likemath Dec 3 '17 at 8:28
• See also this question. – Dietrich Burde Dec 4 '17 at 20:23