Nakayama Lemma - Proof

Proof

A standard proof of the Nakayama lemma uses the following technique due to Atiyah & Macdonald (1969).

• Let M be an R-module generated by n elements, and φ : M → M an R-linear map. If there is an ideal I of R such that φ(M) ⊂ IM, then there is a monic polynomial

with p_k ∈ Ik, such that

as an endomorphism of M.

This assertion is precisely a generalized version of the Cayley–Hamilton theorem, and the proof proceeds along the same lines. On the generators x_i of M, one has a relation of the form

where a_ij ∈ I. Thus

The required result follows by multiplying by the adjugate of the matrix (φδ_ij − a_ij) and invoking Cramer's rule. One finds then det(φδ_ij − a_ij) = 0, so the required polynomial is

To prove Nakayama's lemma from the Cayley–Hamilton theorem, assume that IM = M and take φ to be the identity on M. Then define a polynomial p(x) as above. Then

has the required property.