# Is Hilbert basis theorem true for positive graded ring?

Let $$R=\oplus_{I\geq 0}R_i$$ be a positive graded ring(maybe not commutative), where $$R_0$$ is a commutative Noetherian ring. If $$R$$ is finite generated $$R_0$$-algebra, is $$R$$ Noetherian?

In here, Is every (left) graded-Noetherian graded ring (left) Noetherian?, $$\mathbb Z$$-graded ring is graded Noehterian iff it is Noetherian.

I found that this result is true for graded-commutative ring using Artin-Tate lemma:https://en.wikipedia.org/wiki/Artin-Tate_lemma.

(Translation of the exercise: Let $$K$$ be a commutative field, let $$A$$ be the polynomial ring $$K[T]$$, and let $$\sigma$$ be the endomorphism $$P(T)\mapsto P(T^2)$$ of $$A$$. Then, the ring $$A[X]_\sigma$$ is not left-noetherian, although $$A$$ is noetherian.)
• Done. $\mbox{ }$ – Fred Rohrer Dec 30 '19 at 18:24
Perhaps the simplest counterexample: let $$R$$ be the free $$R_0$$-algebra on two generators $$x$$ and $$y$$. The two-sided ideal $$RxR$$ is not finitely generated as a left (or right) ideal.