Let $R=R_0\oplus R_1 \oplus \ldots$ be a Noetherian graded ring and $M$ a graded $R$-module such that $M$ is finitely generated as an $R$-module. I want to show that for each $n$, $M_n$ is finitely generated as an $R_0$-module.
I have tried two different strategies but with both of them I think I reach dead ends. I have the intuition behind why this result must be true but I am struggling with formalising it.
For the first attempt, I know that there is an isomorphism $\varphi:M\rightarrow \bigoplus_n M_n$. Since $M=R\langle x_1,\ldots x_s\rangle$ for some $x_1,\ldots , x_s\in M$, we must have that $\bigoplus_n M_n=R\langle \varphi(x_1),\ldots \varphi(x_s) \rangle$. We also have the projection maps $$\pi_i: \bigoplus_n M_n \longrightarrow M_i,$$ so I think that $((\pi_n\circ\varphi) (x_j))_{j=1}^s$ would give me a finite set of elements of $M_n$ such that $M_n$ is finitely generated as an $R_0$-module. I am not certain if this is true, however. Specifically, I am certain that this proves that $M_n$ is finitely generated as an $R$-module, but I am not so sure about the assertion that it is finitely generated as an $R_0$-module.
Another approach I have considered would be assuming that $M$ cannot be finitely generated as an $R_0$-module, and then going back along these maps to show that then $M$ cannot be finitely generated as an $R$-module. Again, my problem lies in going from "not finitely generated as an $R_0$-module" to "not finitely generated as an $R$-module". In this case, I would say that if $M_n$ is not finitely generated as an $R_0$-module for some $n$, then $\bigoplus_n M_n$ is certainly not finitely generated as an $R_0$-module, but I don't know how to conclude that it is also not finitely generated as an $R$-module.