 This means that for every pair of elements a Bézout identity holds, and that every finitely generated ideal is principal.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 (which obviously excludes being a PID); if so, it is not a unique factorization domain (UFD), but still is a GCD domain.The theory of Bézout domains retains many of the properties of PIDs, without requiring the Noetherian property.