I call a graded (non-commutative) ring $A$ (left) graded-Noetherian if every homogeneous (left) ideal is finitely generated, and (left) Noetherian if it is (left) Noetherian as a ring.

In the commutative setting, I think I can prove that a graded-Noetherian ring is Noetherian. This basically follows from Hilbert's basis theorem (graded-Noetherianity suffices to show that $A_0$ is Noetherian and that $A$ is finitely generated over $A_0$).

Since Hilbert's basis theorem fails noncommutatively the same line of reasoning will not work in the noncommutative setting.

Q: Is every (left) graded-Noetherian graded ring (left) Noetherian?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.