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> 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?
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In here, [Is every (left) graded-Noetherian graded ring (left) Noetherian?](https://mathoverflow.net/q/303005/106580), $\mathbb Z$-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>.


Thank you in advance.



Is Hilbert basis theorem true for positive graded ring?
The answer is **no** by Exercice 26 in the 2012 edition of Bourbaki's *Algèbre* VIII.1. (This seems moreover to have nothing to do with graduations.)


(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.)


