I call a $\mathbb{Z}$-graded (non-commutative, associative, unital) 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?

I asked this question in MSE a few days ago and got no replies at all, not sure at all where it belongs.