# Nakayama's lemma

From Commalg

This article is about the statement of a simple but indispensable lemma in commutative algebra

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## Statement

Let be a commutative unital ring, and be an ideal contained inside the Jacobson radical of . Let be a finitely generated -module. Then the following are true:

- If then
- If is a submodule of such that , then
- If have images in that generate it as a -module, then generate as a -module

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

## Related facts

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