A UFD such that every prime ideal is contained in a principal proper ideal is a PID?

Asked Aug 21 '16 at 06:41

Active Oct 31 '17 at 21:08

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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.

edited Oct 31 '17 at 21:08

user26857

asked Aug 21 '16 at 06:41

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 – Aug 21 '16 at 07:02
Possible duplicate of If in a UFD every maximal ideal is principal then it is a PID – user26857 Oct 31 '17 at 09:01

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