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 Nakayama (1951), 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 in classic texts by Zariski–Samuel (1958) and Atiyah–Macdonald (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 R {\displaystyle R} be a commutative ring with identity 1. The following is Nakayama's lemma, as stated in Matsumura (1989): Statement 1: Let I {\displaystyle I} be an ideal in R {\displaystyle R} , and M {\displaystyle M} a finitely generated module over R {\displaystyle R} . If I M = M {\displaystyle IM=M} , then there exists r ∈ R {\displaystyle r\in R} with r ≡ 1 ( mod I ) {\displaystyle r\equiv 1\;(\operatorname {mod} I)} such that r M = 0 {\displaystyle rM=0} . This is proven below. A useful mnemonic for Nakayama's lemma is " I M = M ⟹ i m = m {\displaystyle IM=M\implies im=m} ". This summarizes the following alternative formulation: Statement 2: Let I {\displaystyle I} be an ideal in R {\displaystyle R} , and M {\displaystyle M} a finitely generated module over R {\displaystyle R} . If I M = M {\displaystyle IM=M} , then there exists an i ∈ I {\displaystyle i\in I} such that i m = m {\displaystyle im=m} for all m ∈ M {\displaystyle m\in M} .
Proof: Take i = 1 − r {\displaystyle i=1-r} in Statement 1. The following corollary is also known as Nakayama's lemma, and it is in this form that it most often appears. Statement 3: If M {\displaystyle M} is a finitely generated module over R {\displaystyle R} , J ( R ) {\displaystyle \mathrm {J} (R)} is the Jacobson radical of R {\displaystyle R} , and J ( R ) M = M {\displaystyle \mathrm {J} (R)M=M} , then M = 0 {\displaystyle M=0} .
Proof: 1 − r {\displaystyle 1-r} (with r {\displaystyle r} as in Statement 1) is in the Jacobson radical so r {\displaystyle r} is invertible. More generally, one has that J ( R ) M {\displaystyle \mathrm {J} (R)M} is a superfluous submodule of M {\displaystyle M} when M {\displaystyle M} is finitely generated. Statement 4: If M {\displaystyle M} is a finitely generated module over R {\displaystyle R} , N {\displaystyle N} is a submodule of M {\displaystyle M} , and M = N + J ( R ) M {\displaystyle M=N+\mathrm {J} (R)M} , then M = N {\displaystyle M=N} .
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