Nakayama's lemma

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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.