Exercise 13, § 1

Let A be a Noetherian ring, E a finitely generated A-module, p, q two prime ideals of A such that $p \subset q$ and a an element of q. Suppose that $p \in \operatorname{Ass}(E)$ and that the homothety $a_E$ is injective. Show that there exists a prime ideal $n \in \operatorname{Ass}(E/aE)$ such that $p + Aa \subset n \subset q$ (Replacing A by A_q, we may assume that A is local with maximal ideal q. Let F $\neq 0$ be the submodule of E consisting of the x such that $px = 0$. Show that the relation $F \subset aE$ would imply $F = aF$ and obtain a contradiction using Nakayama's Lemma. Then use Proposition 8 of no. 4.)