Nakayama's lemma: Difference between revisions

Latest revision as of 16:27, 12 May 2008

This article is about the statement of a simple but indispensable lemma in commutative algebra
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Statement

Let $R$ be a commutative unital ring, and $I$ be an ideal contained inside the Jacobson radical of $R$ . Let $M$ be a finitely generated $R$ -module. Then the following are true:

If $I M = M$ then $M = 0$
If $N$ is a submodule of $M$ such that $N + I M = M$ , then $N = M$
If $m_{1}, m_{2}, \dots, m_{n}$ have images in $M / I M$ that generate it as a $R$ -module, then $m_{1}, m_{2}, \dots, m_{n}$ generate $M$ as a $R$ -module

In the particular case where $R$ is a local ring, the Jacobson radical is the unique maximal ideal in $R$ .

Related facts

The graded Nakayama's lemma is a related fact true for graded rings.

@@ Line 9: / Line 9: @@
 # If <math>m_1, m_2, \ldots, m_n</math> have images in <math>M/IM</math> that generate it as a <math>R</math>-module, then <math>m_1, m_2, \ldots, m_n</math> generate <math>M</math> as a <math>R</math>-module
+In the particular case where <math>R</math> is a [[local ring]], the Jacobson radical is the unique [[maximal ideal]] in <math>R</math>.
 ==Related facts==
 The [[graded Nakayama's lemma]] is a related fact true for [[graded ring]]s.