In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative ring behave like vector spaces over a field. It is an important tool in algebraic geometry, because it allows local data on algebraic varieties, in the form of modules over local rings, to be studied pointwise as vector spaces over the residue field of the ring. The lemma is named after the Japanese mathematician Tadashi Nakayama and introduced in its present form in , although it was first discovered in the special case of ideals in a commutative ring by Wolfgang Krull and then in general by Goro Azumaya (1951). In the commutative case, the lemma is a simple consequence of a generalized form of the Cayley–Hamilton theorem, an observation made by Michael Atiyah (1969). The special case of the noncommutative version of the lemma for right ideals appears in Nathan Jacobson (1945), and so the noncommutative Nakayama lemma is sometimes known as the Jacobson–Azumaya theorem. The latter has various applications in the theory of Jacobson radicals. Statement Let be a commutative ring with identity 1. The following is Nakayama's lemma, as stated in : Statement 1: Let be an ideal in , and a finitely-generated module over . If , then there exists an with , such that . 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: If is a finitely-generated module over , is the Jacobson radical of , and , then . Proof: (with as above) is in the Jacobson radical so is invertible. More generally, one has that is a superfluous submodule of when is finitely-generated. Statement 3: If is a finitely-generated module over R, N is a submodule of , and M = N + J(R)M, then M = N. Proof: Apply Statement 2 to M/N. The following result manifests Nakayama's lemma in terms of generators. Statement 4: If M is a finitely-generated module over R and the images of elements m1,...,mn of M in M / J(R)M generate M / J(R)M as an R-module, then m1,...,mn also generate M as an R-module.