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:


 * 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.