A theorem of Hungerford says that : Every PIR (principal ideal ring , obviously commutative ) is a homomorphic image of a finite direct product of PID s . My question is , is there a similar criteria for Noetherian rings, i.e. : Is every Noetherian ring a homomorphic image of a finite direct product of Noetherian domains ?
This Is every Noetherian Commutative Ring a quotient of a Noetherian Domain? shows that we cannot expect every Noetherian ring to be a homomorphic image of a Noetherian domain.