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

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+{{indispensable lemma}}
 ==Statement==
 Let <math>R</math> be a [[commutative unital ring]], and <math>I</math> be an [[ideal]] contained inside the [[Jacobson radical]] of <math>R</math>. Let <math>M</math> be a finitely generated <math>R</math>-module. Then the following are true:
-* If <math>IM = M</math> then <math>M = 0</math>
+# If <math>IM = M</math> then <math>M = 0</math>
-* 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
+# If <math>N</math> is a submodule of <math>M</math> such that <math>N + IM = M</math>, then <math>N = M</math>
+# 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.