Nakayama's Lemma states that if M is a finitely generated module over a commutative ring and the ring's Jacobson radical, multiplied into M, equals M itself, then M must be the zero module. Named for Tadashi Nakayama, it is a central tool of commutative algebra, most often used to show that a proposed set of elements generates a module over a local ring by checking only that their images generate the simpler quotient by the maximal ideal.
Facts
StatementNakayama's lemma states that if I is an ideal in a commutative ring R and M is a finitely generated R-module with IM equal to M, then there exists an element r of R congruent to 1 modulo I such that rM is zero; it governs how the Jacobson radical of R controls finitely generated modules over R. 1 Classification
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Sources
1. Nakayama's Lemma (Wikipedia)
Wikimedia Foundationlead section, first paragraph
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.
lead section, second paragraph
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).
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