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Are there any integral domains in which no nonzero prime ideal is finitely generated? (Other than fields, of course, where the condition is vacuously satisfied.)

I asked a similar question the other day, but the solution there relied on using zero-divisors and that didn't really help clear up the situation I was considering.

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Daniel McLaury
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  • The similar question is about commutative rings with no nonzero finitely generated prime ideal: see http://math.stackexchange.com/questions/915615/are-there-any-commutative-rings-in-which-no-nonzero-prime-ideal-is-finitely-gene – Dietrich Burde Sep 02 '14 at 09:20

1 Answers1

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Every valuation ring of rank one which is not discrete satisfy your requirement.

A concrete example you can find here. Another one is the integral closure of $\mathbb Z_p$ (the ring of $p$-adic integers) in $\overline{\mathbb Q}_p$ (the algebraic closure of $\mathbb Q_p$, the field of $p$-adic numbers).

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user26857
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