Exercise 6, § 1

Let A be a ring and E an A-module.

(a) Show that for a prime ideal p of A to belong to Supp(E), it is necessary and sufficient that there exist a submodule F of E such that $p \in \operatorname{Ass}(E/F)$ (to see that the condition is necessary, consider a submodule F of E of the form px, where $x \in E$; to see that the condition is sufficient, use Proposition 7 of no. 3).

(b) Suppose that A is Noetherian and E finitely generated. Show that for every prime ideal $p \in \operatorname{Supp}(E)$, there exists a composition series $(E_i)_{0 \leq i \leq n}$ of E such that, for $0 \leq i \leq n - 1$, $E_i/E_{i+1}$ is isomorphic to $A/p_i$, where $p_i$ is a prime ideal and one of the $p_i$ is equal to p.