This theorem is to show the uniqueness of the Structure Theorem
- Can someone elaborate the part where they deduce the $\dim M(p)?$
For context let me elaborate the previous sentence preceding the dimension argument. Define $R/(a_i) \stackrel{\phi_i}\to M(p)$ where $\phi_i(x_i) = px_i.$ Then the decomposition mentioned is $M(p) = \oplus_{i =1}^t \ker(\phi_i).$ Now I feel like I know the dimension of the kernel, I can figure this out
EDIT1 I think there is a typo. By $x_i \in R/(a_i)$ they probably meant $\bar{x}_i \in R/(q_i)$. So by an abuse of notation we should let $\phi(\bar(x)_i) = p\bar{x}_i.$ The kernel is then $\{\bar{x}_i : p\bar{x_i} \in R/(q_i) \} = \{px_i \in (q_i) \}.$ Now this prove (1).
- I just need some elaboration on the decomposition (mainly the last equality). It looks obvious but I feel like i need some details or at least hints
$$pM = \oplus pR/(pb_i) = \oplus R/(b_i)$$