TITLE: On Prufer domains
QUESTION [5 upvotes]: Are there any Prufer domains that have an infinity of prime ideals but only one of those primes is not finitely generated?

REPLY [2 votes]: In $\mathbb{Z}+X\mathbb{Q}[X]$, all prime ideals are principal except $X\mathbb{Q}[X]$ which is not finitely generated. Also $\mathbb{Z}+X\mathbb{Q}[X]$ is a Bezout domain (A domain in which each finitely generated ideal is principal) and a Bezout domain is always Prufer.