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Let $D$ be a UFD such that every prime ideal is contained in a principal proper ideal. Is then $D$ a PID ?

I can only prove that if every proper ideal of a UFD is contained in a principal proper ideal then it is a PID. Please help. Thanks in advance.

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  • $\begingroup$ Now I reflect on it , it seems trivial by the result I already know ... every proper ideal is contained in a prime ideal and by assumption it is contained in a principal proper ideal ... so done $\endgroup$ – user228168 Aug 21 '16 at 7:02
  • $\begingroup$ Possible duplicate of If in a UFD every maximal ideal is principal then it is a PID $\endgroup$ – user26857 Oct 31 '17 at 9:01

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