Finitely generated ideal - Revision history

Vipul: 3 revisions

2008-05-12T16:21:50Z

3 revisions

← Older revision	Revision as of 16:21, 12 May 2008
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Vipul: /* Stronger properties */

2008-01-11T16:20:51Z

Stronger properties

← Older revision		Revision as of 16:20, 11 January 2008
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	* [[Principal ideal]]		* [[Principal ideal]]
			* [[Noetherian ideal]]

	==Metaproperties==		==Metaproperties==

Vipul at 16:20, 11 January 2008

2008-01-11T16:20:13Z

← Older revision		Revision as of 16:20, 11 January 2008
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	==Definition ~~for commutative rings~~==		{{curing-ideal property}}

			{{ideal-as-a-module\|finitely generated module}}

			==Definition==

	===Symbol-free definition===		===Symbol-free definition===
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	An [[ideal]] <math>I</math> in a [[commutative unital ring]] <math>R</math> is said to be '''finitely generated''' if there is a finite set <math>x_1, x_2, ..., x_n</math> such that <math>I = Rx_1 + Rx_2 + ... + Rx_n</math>.		An [[ideal]] <math>I</math> in a [[commutative unital ring]] <math>R</math> is said to be '''finitely generated''' if there is a finite set <math>x_1, x_2, ..., x_n</math> such that <math>I = Rx_1 + Rx_2 + ... + Rx_n</math>.

	==~~Definition for noncommutative rings~~==		==Relation with other properties==

			===Stronger properties===

	==~~=Symbol-free definition=~~==		* [[Principal ideal]]

			==Metaproperties==

	~~An [[~~ideal]] in a [[commutative unital ring]] is said to be '''finitely generated''' if it has a finite generating set, that is, if there is a finite set such that it is the smallest ideal containing that finite set.		{{trim ideal property}}

	~~===Definition with symbols===~~		{{finite-sum-closed ideal property}}

	{{~~fillin~~}}		{{not intersection-closed ideal property}}

	[[~~Category: Properties of ideals in commutative rings~~]]		In general, an intersection of finitely generated ideals need not be finitely generated. However, for [[Noetherian ring]]s, where every ideal is finitely generated, an intersection of finitely generated ideals is certainly finitely generated.
	~~[[Category: Properties~~ of ideals ~~in noncommutative rings]]~~

Vipul: Started the page

2007-01-09T04:51:20Z

Started the page

New page

==Definition for commutative rings==

===Symbol-free definition===

An [[ideal]] in a [[commutative unital ring]] is said to be '''finitely generated''' if it has a finite generating set, that is, if there is a finite set such that it is the smallest ideal containing that finite set.

===Definition with symbols===

An [[ideal]] <math>I</math> in a [[commutative unital ring]] <math>R</math> is said to be '''finitely generated''' if there is a finite set <math>x_1, x_2, ..., x_n</math> such that <math>I = Rx_1 + Rx_2 + ... + Rx_n</math>.

==Definition for noncommutative rings==

===Symbol-free definition===

An [[ideal]] in a [[commutative unital ring]] is said to be '''finitely generated''' if it has a finite generating set, that is, if there is a finite set such that it is the smallest ideal containing that finite set.

===Definition with symbols===

{{fillin}}

[[Category: Properties of ideals in commutative rings]]
[[Category: Properties of ideals in noncommutative rings]]