Exercise 10, § 2

Let $A$ be a Noetherian ring and $M$ an $A$-module. Show that the following properties are equivalent:

(α) Every finitely generated submodule of $M$ is of finite length. (β) $M$ is a direct limit of $A$-modules of finite length. (γ) Every element $p \in \operatorname{Ass}(M)$ is a maximal ideal of $A$. (δ) Every element $p \in \operatorname{Supp}(M)$ is a maximal ideal of $A$.