Nakayama's Lemma

Theorem
Let $A$ be a commutative ring with unity.

Let $M$ be a finitely generated $A$-module.

Let $\map {\operatorname{Jac} } A$ be the Jacobson radical of $A$.

Let $\mathfrak a \subseteq \map {\operatorname{Jac} } A$ be an ideal of $A$.

Suppose $\mathfrak a M = M$.

Then:
 * $M = 0$

Also see

 * Cayley-Hamilton Theorem