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 $x \prec y$.