High Quality Content by WIKIPEDIA articles! In mathematics, more specifically abstract algebra and especially 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 a significant 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.

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