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- Nakayama_lemma abstract "In mathematics, more specifically modern 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 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.".
- Nakayama_lemma wikiPageID "1835001".
- Nakayama_lemma wikiPageRevisionID "606071320".
- Nakayama_lemma hasPhotoCollection Nakayama_lemma.
- Nakayama_lemma subject Category:Algebraic_geometry.
- Nakayama_lemma subject Category:Commutative_algebra.
- Nakayama_lemma subject Category:Lemmas.
- Nakayama_lemma subject Category:Theorems_in_abstract_algebra.
- Nakayama_lemma type Abstraction100002137.
- Nakayama_lemma type Communication100033020.
- Nakayama_lemma type Lemma106751833.
- Nakayama_lemma type Lemmas.
- Nakayama_lemma type Message106598915.
- Nakayama_lemma type Proposition106750804.
- Nakayama_lemma type Statement106722453.
- Nakayama_lemma type Theorem106752293.
- Nakayama_lemma type TheoremsInAbstractAlgebra.
- Nakayama_lemma comment "In mathematics, more specifically modern 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.".
- Nakayama_lemma label "Lemma di Nakayama".
- Nakayama_lemma label "Lemma van Nakayama".
- Nakayama_lemma label "Lemma von Nakayama".
- Nakayama_lemma label "Lemme de Nakayama".
- Nakayama_lemma label "Nakayama lemma".
- Nakayama_lemma label "Лемма Накаямы".
- Nakayama_lemma label "中山引理".
- Nakayama_lemma sameAs Lemma_von_Nakayama.
- Nakayama_lemma sameAs Lemme_de_Nakayama.
- Nakayama_lemma sameAs Lemma_di_Nakayama.
- Nakayama_lemma sameAs Lemma_van_Nakayama.
- Nakayama_lemma sameAs m.05_q8x.
- Nakayama_lemma sameAs Q1399751.
- Nakayama_lemma sameAs Q1399751.
- Nakayama_lemma sameAs Nakayama_lemma.
- Nakayama_lemma wasDerivedFrom Nakayama_lemma?oldid=606071320.
- Nakayama_lemma isPrimaryTopicOf Nakayama_lemma.