I quoted Theorem 13.2 below from Matsumura's book Commutative Ring Theory:
Let $R=\bigoplus_{n\geq 0}R_n$ be a Noetherian graded ring with $R_0$ Artinian, and let M be a finitely generated graded $R$-module. Suppose that $R=R_0[x_1,...,x_r]$ with $x_i$ of degree $d_i$, and that $$P(M,t)=\sum_{n=0}^{\infty}l(M_n)t^n\in\Bbb Z\left[\left[ t\right]\right]$$ Then $P(M, t)$ is a rational function of $t$, and can be written $$P(M,t)=\frac{f(t)}{\prod_{i=1}^r(1-t^{d_i})}.$$
I don't see how the Artinian assumption of $R_0$ is necessary here. If one starts from the initial inductive assumption $n=0$ and $R=R_0\oplus 0...$ then still $M_n$ will eventually vanish at large $n$ as $M$ is finitely generated and only finite many of $M_n$ suffices to generate $M$. Maybe the author was trying to argue this differently by looking at the following decreasing chain:
$$M_1\oplus M_2...\supset M_2\oplus M_3...\supset...$$
and use the assumption on the Artinian module $M$ that the above chain will eventually become stable?
How is Artinian assumption necessary here?