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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.

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

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