Any principal ideal domain (PID) is a Bézout domain, but a Bézout domain need not be a Noetherian ring, so it could have non-finitely generated ideals; if so, it is not a unique factorization domain (UFD), but is still a GCD domain. The theory of Bézout domains retains many of the properties of PIDs, without requiring the Noetherian property.

Bézout domains