I have a question about this statement:
Let $M$ be an $R$-module of finite length and let $\mbox{Ann}(M)\subset P_1,...,P_k$ be maximal ideals. If $n\in\mathbb{Z}_+$ is such that $P_1^n\cdots P_k^n\subset\mbox{Ann}(M)$, then the sequences $0\rightarrow P_i^nM \rightarrow M \rightarrow M_{P_i} \rightarrow 0$ are exact and $M/P_iM \cong M_{P_i}$.
I wonder what the maps are in these sequences. Especially what is the homomorphism $M \rightarrow M_{P_i}$? I especially wonder because I can not find such a surjective homomorphism.