Nakayama Lemma - Statement

Statement

Let R be a commutative ring with identity 1. The following is Nakayama's lemma, as stated in Matsumura (1989):

Statement 1: Let I be an ideal in R, and M a finitely-generated module over R. If IM = M, then there exists an r ∈ R with r ≡ 1 (mod I), such that rM = 0.

This is proven below.

The following corollary is also known as Nakayama's lemma, and it is in this form that it most often appears.

Statement 2: With conditions as above, if I is contained in the Jacobson radical of R, then necessarily M = 0.

Proof: r−1 (with r as above) is in the Jacobson radical so r is invertible.

More generally, one has

Statement 3: If M = N + IM for some ideal I in the Jacobson radical of R and M is finitely-generated, then M = N.

Proof: Apply Statement 2 to M/N.

The following result manifests Nakayama's lemma in terms of generators

Statement 4: Let I be an ideal in the Jacobson radical of R, and suppose that M is finitely-generated. If m₁,...,m_n have images in M/IM that generate it as an R-module, then m₁,...,m_n also generate M as an R-module.

Proof: Apply Statement 2 to N = M/Σ_iRm_i.

This conclusion of the last corollary holds without assuming in advance that M is finitely generated, provided that M is assumed to be a complete and separated module with respect to the I-adic topology. Here separatedness means that the I-adic topology satisfies the T₁ separation axiom, and is equivalent to