Nakayama's lemma

Nakayama's lemma states that if a finitely generated module over a local ring equals its product with the ring’s maximal ideal, the module is zero. It also yields generator and surjectivity criteria for modules over local rings.

Local ring: A commutative ring with exactly one maximal ideal. The theorem’s local form uses this unique maximal ideal to force a finite module to vanish.
Module: An abelian group equipped with scalar multiplication by elements of a ring. Nakayama’s lemma concerns modules and how ideals act on them.
Minimal generating set: A generating set from which no element can be removed while preserving generation. Modulo the maximal ideal, Nakayama’s lemma identifies the size of a minimal generating set.
Krull intersection theorem: A theorem describing the intersection of all powers of an ideal in a Noetherian ring or module. It studies repeated ideal multiplication rather than a single equality with the maximal-ideal multiple.
Maximal ideal: A proper ideal maximal among the proper ideals of a ring. Multiplication by this ideal is the operation whose fixed finite modules Nakayama’s lemma constrains.
Module quotient: A module formed by identifying elements whose difference lies in a submodule. The generator criterion examines a module after quotienting by its ideal multiple.
Cotangent space: For a local ring with maximal ideal m and residue field k, the vector space m/m². Its dimension measures the minimal number of generators of the maximal ideal through Nakayama’s lemma.
Cramer's rule: A formula expressing solutions of an invertible linear system using determinants. Both arguments use determinants, but Cramer’s rule solves systems while Nakayama’s trick proves module vanishing.
Finitely generated module: A module generated by a finite set of elements. Finite generation is the hypothesis that makes the lemma’s vanishing conclusion possible.
Ideal product: The submodule generated by products of elements from an ideal and a module. The theorem compares a module directly with its product by the maximal ideal.

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