Let $R=R_0\oplus R_1\oplus R_2\oplus\cdots$ be a graded ring. If $R$ is Noetherian, then I can see that $R_0$ is Noetherian and $R_1\oplus R_2\oplus\cdots$ is a finitely generated ideal of $R.$
But, is it also true that, $R$ is also a finitely generated $R_0$-module?
Edit: As mentioned by Alekos, it's not true. Then, can we say that $R$ is a finitely generated $R_0$-algebra?