I am trying to understand the proof of the following Theorem from Atiyah-MacDonald.
$P(M,t)$ is a rational function in t of the form $f(t)/\prod_{i=1}^{s}(1-t^{k_{i}})$
$P(M,t)=\sum_{n=0}\lambda(M_{n})t^{n}$ is the Poincare Series and $\lambda$ is an additive function on a graded module $M$.
Here we are dealing with a Noetherian graded ring $A=\bigoplus_{n=1}^{\infty} A_{n}$ where $A$ can be viewed as a fintiely generated $A_{0}$-algebra, that is $A=A_{0}[x_{1}, \ldots, x_{s}]$ where we have $x_{i} \in A_{k_{i}}$. This is the meaning of the $k_{i}$.
The proof goes by induction of the number of generators of $A$ as an $A_{0}$ algebra. The base case occurs when $A=A_{0}$.
I was trying to understand the following simple sub-case of the base case.
Suppose $M$ is generated by $m_{2}$ in $M_{2}$ then as consequences of the definition of graded rings and modules we have that $M_{i}=0$ for $i \neq 2$. In which case we have
$P(M,t)=\lambda(M_{2})t^{2}+\lambda(0)(1+t+t^{3} + \cdots)$
My question is essentially can we assume that $\lambda(0)=0$ to justify the proof?
