Example 11.3 in Atiyah Macdonald book

Question

Look at this problem: Is the length of the composition series of a free module identical to the number of its bases?

I wanna confirm that the example is false or correct. Moreover, if it is incorrect, is $P(M,t)=l(A_{0})(1-t)^{-s}$ right? because of $l(A_{n})=l(A_{0})\binom{n+s-1}{s-1}$ (*).

Can you prove (*)? This equation is still true if $A_{0}$ is not Artin in this case.

Answer 1

I give the answer to close this topic.

$R_{n}=\oplus_{\sum_{m_{i}}=n} (R_{0}x_{1}^{m_{1}}...x_{r}^{m_{r}})\Rightarrow l(R_{n})=\oplus_{\sum_{m_{i}}=n} l(R_{0}x_{1}^{m_{1}}...x_{r}^{m_{r}})$

$\Rightarrow l(R_{n})=l(R_{0})|x_{1}^{m_{1}}...x_{r}^{m_{r}}|_{\sum_{m_{i}}=n}=l(R_{0})\binom{n+r-1}{r-1}$