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:

1. If $IM=M$ then $M=0$
2. If $N$ is a submodule of $M$ such that $N+IM=M$, then $N=M$
3. 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

In the particular case where $R$ is a local ring, the Jacobson radical is the unique maximal ideal in $R$.

Related facts

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