Exercise 1, § 8

Let $A$ be a local ring noetherian, $M$ an a-module. Now prove that for $M$ to be artinian, it is necessary and sufficient that its socle $S$ be of finite dimension over $\kappa_A$ and that $M$ be an essential extension of $S$ (that is to say (A, II, p. 185, exerc. 15) that every nonzero submodule of $M$ contain a nonzero element of $S$).