Nakayama's lemma

Nakayama's lemma is a fundamental result in commutative algebra that provides conditions under which a module over a ring can be simplified. It is particularly useful in the study of finitely generated modules over local rings or Noetherian rings. The classic statement of Nakayama's lemma can be summarized as follows: Let \( R \) be a Noetherian ring, and let \( M \) be a finitely generated \( R \)-module.