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