Nakayama's lemma

This article is about the statement of a simple but indispensable lemma in commutative algebra
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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 $N$ is a submodule of $M$ suc hthat $N + I M = M$ , then $N = M$
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

Related facts

The graded Nakayama's lemma is a related fact true for graded rings.