Nakayama's lemma - Revision history

Vipul: 5 revisions

2008-05-12T16:27:35Z

5 revisions

← Older revision	Revision as of 16:27, 12 May 2008
(No difference)

Vipul at 17:52, 3 March 2008

2008-03-03T17:52:27Z

← Older revision		Revision as of 17:52, 3 March 2008
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	# 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>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==		==Related facts==

	The [[graded Nakayama's lemma]] is a related fact true for [[graded ring]]s.		The [[graded Nakayama's lemma]] is a related fact true for [[graded ring]]s.

Vipul: /* Statement */

2008-03-03T17:44:57Z

Statement

← Older revision		Revision as of 17:44, 3 March 2008
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	# If <math>IM = M</math> then <math>M = 0</math>		# If <math>IM = M</math> then <math>M = 0</math>
	# If <math>N</math> is a submodule of <math>M</math> ~~suc hthat~~ <math>N + IM = M</math>, then <math>N = M</math>		# 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		# 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

Vipul: /* Statement */

2008-03-03T17:44:43Z

Statement

← Older revision		Revision as of 17:44, 3 March 2008
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	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:		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> suc hthat <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

			==Related facts==

			The [[graded Nakayama's lemma]] is a related fact true for [[graded ring]]s.

Vipul at 21:18, 8 February 2008

2008-02-08T21:18:26Z

← Older revision		Revision as of 21:18, 8 February 2008
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			{{indispensable lemma}}

	==Statement==		==Statement==

Vipul at 11:33, 7 August 2007

2007-08-07T11:33:19Z

New page

==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>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