Artinian ring - Revision history

Vipul: 8 revisions

2008-05-12T16:18:44Z

8 revisions

← Older revision	Revision as of 16:18, 12 May 2008
(No difference)

Vipul at 01:00, 19 February 2008

2008-02-19T01:00:43Z

← Older revision		Revision as of 01:00, 19 February 2008
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			{{semibasicdef}}
	{{curing property}}		{{curing property}}

Vipul at 17:19, 11 January 2008

2008-01-11T17:19:30Z

← Older revision		Revision as of 17:19, 11 January 2008
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	{{curing property}}		{{curing property}}

			{{domain-to-field property}}

	==Definition==		==Definition==

Vipul: /* Weaker properties */

2008-01-10T01:18:42Z

Weaker properties

← Older revision		Revision as of 01:18, 10 January 2008
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	===Weaker properties===		===Weaker properties===

	* [[Noetherian ring]]		* [[Noetherian ring]]: {{proofat\|[[Artinian implies Noetherian]]}}
	* [[Zero-dimensional ring]]		* [[Zero-dimensional ring]]: {{proofat\|[[Artinian implies zero-dimensional]]}}
	* [[Jacobson ring]]		* [[Jacobson ring]]: {{proofat\|[[Artinian implies Jacobson]]}}

Vipul: /* Weaker properties */

2008-01-10T01:15:51Z

Weaker properties

← Older revision		Revision as of 01:15, 10 January 2008
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	* [[Noetherian ring]]		* [[Noetherian ring]]
	* [[Zero-dimensional ring]]		* [[Zero-dimensional ring]]
			* [[Jacobson ring]]

Vipul at 09:10, 7 August 2007

2007-08-07T09:10:39Z

← Older revision		Revision as of 09:10, 7 August 2007
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	{{~~commring~~ property}}		{{curing property}}

	==Definition==		==Definition==

Vipul at 15:36, 30 June 2007

2007-06-30T15:36:44Z

← Older revision		Revision as of 15:36, 30 June 2007
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	==Definition ~~for commutative rings~~==		{{commring property}}

			==Definition==

	===Symbol-free definition===		===Symbol-free definition===

	A [[commutative unital ring]] ~~(or more generally a [[commutative ring]])~~ is termed '''Artinian''' if it satisfies the [[descending chain condition]] on [[ideal]]s, that is, any descending chain of ideals stabilizes after a finite length.		A [[commutative unital ring]] is termed '''Artinian''' if it satisfies the [[descending chain condition]] on [[ideal]]s, that is, any descending chain of ideals stabilizes after a finite length.

			==Relation with other properties==

			===Stronger properties===

			* [[Field]]

	===~~Definition with symbols~~===		===Weaker properties===

	~~{{fillin}}~~		* [[Noetherian ring]]
			* [[Zero-dimensional ring]]

Vipul: /* Definition for non-commutative rings */

2007-06-30T15:31:48Z

Definition for non-commutative rings

← Older revision		Revision as of 15:31, 30 June 2007
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	{{fillin}}		{{fillin}}

	~~==Definition for non-commutative rings==~~

	~~There are two notions:~~

	* [[Left Artinian ring]] must satisfy the descending chain condition on [[left ideal]]s
	* [[Right Artinian ring]] must satisfy the descending chain condition on [[right ideal]]s

	~~[[Category: Properties of commutative rings]]~~

Vipul: Started the page

2007-01-09T02:21:19Z

Started the page

New page

==Definition for commutative rings==

===Symbol-free definition===

A [[commutative unital ring]] (or more generally a [[commutative ring]]) is termed '''Artinian''' if it satisfies the [[descending chain condition]] on [[ideal]]s, that is, any descending chain of ideals stabilizes after a finite length.

===Definition with symbols===

{{fillin}}

==Definition for non-commutative rings==

There are two notions:

* [[Left Artinian ring]] must satisfy the descending chain condition on [[left ideal]]s
* [[Right Artinian ring]] must satisfy the descending chain condition on [[right ideal]]s

[[Category: Properties of commutative rings]]