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Nakayama's lemma (Theorem)

Let $ R$ be a commutative ring with 1. Let $ M$ be a finitely generated $ R$-module. If there exists an ideal $ \mathfrak{a}$ of $ R$ contained in the Jacobson radical and such that $ \mathfrak{a}M = M$, then $ M=0$.



"Nakayama's lemma" is owned by n3o.
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Attachments:
proof of Nakayama's lemma (Proof) by mclase
proof of Nakayama's lemma (Proof) by nerdy2
counter example to Nakayama's lemma for non-finitely generated modules (Example) by sjm
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Cross-references: Jacobson radical, contained, ideal, finitely generated, commutative ring
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This is version 3 of Nakayama's lemma, born on 2002-11-03, modified 2003-10-04.
Object id is 3563, canonical name is NakayamasLemma.
Accessed 5435 times total.

Classification:
AMS MSC: 13C99 (Commutative rings and algebras :: Theory of modules and ideals :: Miscellaneous)
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