What is an example for a GCD domain that is neither a UFD (like $\Bbb Z[X]$) nor Bézout (like holomorphic functions on all of $\Bbb C$)?
The confluence of rings being a Bézout domain AND a UFD are all the PIDs. This is an equivalence. The two examples given are GCD's that are either not Bézout, or not UFD. But there are some GCD's that are neither of the two, I just could not find one easily, apart from: It seems polynomials over a non-PID Bézout ring would fit the bill, e.g. over the ring of all algebraic intergers (that doesn't satisfy the ACCP), but does anyone have a simpler example?