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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