I am working on exercise 3.12 from Eisenbud's Commutative Algebra and I am having trouble parsing the question.
Let $M$ be a finitely generated module over the Noetherian ring $R$. Given any multiplicatively closed set $U \subset R$, show that the intersection of the primary components of $0$ in $M$ corresponding to those primes of Ass $M$ not meeting $U$ is the kernel of the localization map $M\to M[U^{-1}]$, and is thus independent of the primary decomposition chosen.
I am just looking for a hint to get started. Also, if you have the book, what are the relevant theorems for solving this question?
For $s\in U$ and $x\in M$ we have $sx=0\implies sx\in Q$, where $Q$ is a primary submodule of $M$ which appears in a primary decomposition of $(0)$ and $r_M(Q)\cap U=\emptyset$. Conclude that $x\in Q$.
For the converse, $x\in\cap Q_j$ with $r_M(Q_j)\cap U=\emptyset$. On the other side, for some $Q_i$ such that $r_M(Q_i)\cap U\ne\emptyset$ we get an $s_i\in r_M(Q_i)\cap U$, so $s_i^{k_i}M\subseteq Q_i$. In particular, $s_i^{k_i}x\in Q_i$. Now find an $s\in U$ such that $sx=0$.