Nakayama's lemma

Statement

Let $R$ be a commutative unital ring, and $I$ be an ideal contained inside the Jacobson radical of $R$ . Let $M$ be a finitely generated $R$ -module. Then the following are true:

If $I M = M$ then $M = 0$
If $m_{1}, m_{2}, \dots, m_{n}$ have images in $M / I M$ that generate it as a $R$ -module, then $m_{1}, m_{2}, \dots, m_{n}$ generate $M$ as a $R$ -module