I am seeing Jean Pierre Serre's book entitled "local algebra."
At the beginning of chapter 1 they state the motto of Nakayama. I will change some notations and place what the book says
Proposition 1. Let $M$ be a finitely generated $A$ -module, and $I$ be an ideal of $A$ contained in the Jacobson radical $J$ of $A$. If $IM = M$ , then $M =0$ .
Proof:
Indeed, if $M\neq 0$ , it has a quotient which is a simple module (because if $M$ is finely generated then there is a submodule $N$ of $M$ such that the quotient $M/N$ is simple), hence is isomorphic to $A/m$, where $m$ is a maximal ideal of $A$; then $mM\neq M$ , contrary to the fact that $I\subset J$.
Why $mM\neq M$ ?