I have been able to show that for a graded subring $S$ of $R[x]$ (where $R$ is a noetherian domain) that in order to show that $S$ is noetherian, it suffices to consider homogeneous ideals of $S$. This proof was very similar to that of Hilbert's basis theorem.

I'm now considering graded subrings in $R[x,y]$ and I would like to prove a similar result to the one above, but this time that it suffices to consider homogeneous ideals in this new $\mathbb{N}^2$ grading. Does anyone have any suggestions? I'm hoping it will follow by an induction but I don't see how at the moment.

I should mention that the grading I'm using on $R[x]$ is the standard one with $R$ in degree $0$ and $x$ in degree $1$. For $R[x,y]$ I would use $R$ in degree $(0,0)$, $x$ in degree $(1,0)$ and $y$ in degree $(0,1)$ with $x \prec y$.


This follows from a general result about coarsening of graded noetherianity.

Let $G$ be an abelian group, and let $R$ be a $G$-graded ring. Then, the following statements are equivalent: (i) $G$ is of finite type; (ii) If $\psi\colon G\rightarrow H$ is an epimorphism of abelian groups and the $G$-graded ring $R$ is noetherian, then the $H$-graded ring $R_{[\psi]}$ is noetherian.

Here, a noetherianity of a graded ring means that every homogeneous ideal is of finite type. Moreover, the coarsening $R_{[\psi]}$ of a $G$-graded ring $R=\bigoplus_{g\in G}R_g$ with respect to an epimorphism of abelian groups $\psi\colon G\rightarrow H$ is the $H$-graded ring whose underlying ring is the ring underlying $R$ and whose graded component of degree $h$ is $\bigoplus_{g\in\psi^{-1}(h)}R_g$ for every $h\in H$.

For a proof, see Theorem 1.1 in this article and Remark 2.14 in this article.

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