Local ring - Revision history

Vipul at 16:37, 20 January 2009

2009-01-20T16:37:34Z

← Older revision		Revision as of 16:37, 20 January 2009
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	==Definition for noncommutative rings==		==Definition for noncommutative rings==

	{{further\|[[Local ring ~~(noncommutative rings~~)]]}}		{{further\|[[Noncommutative:Local ring)]]}}
	==Relation with other properties==		==Relation with other properties==

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	* [[Local domain]]		* [[Local domain]]

	[[~~Category~~: ~~Properties of commutative rings~~]]		===Analogues in other algebraic structures===

			* [[Groupprops:One-headed group\|One-headed group]] in group theory, is a group that has a proper normal subgroup that contains every proper normal subgroup.
			* [[Noncommutative:Local ring\|Local ring]] in noncommutative algebra.

Vipul: 5 revisions

2008-05-12T16:26:36Z

5 revisions

← Older revision	Revision as of 16:26, 12 May 2008
(No difference)

Vipul: /* Stronger properties */

2008-01-20T21:49:15Z

Stronger properties

← Older revision		Revision as of 21:49, 20 January 2008
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	* [[Regular local ring]]		* [[Regular local ring]]
	* [[Completely primary ring]]		* [[Completely primary ring]]
			* [[Local Artinian ring]]

	===Weaker properties===		===Weaker properties===

Vipul at 21:48, 20 January 2008

2008-01-20T21:48:51Z

← Older revision		Revision as of 21:48, 20 January 2008
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	{{curing property}}		{{curing property}}

	==Definition ~~for commutative rings~~==		{{spectrum-determined curing property\|topological space with exactly one closed point}}

			==Definition==

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

	A [[commutative unital ring]] is termed a '''local ring''' if it has a unique [[maximal ideal]].		A [[commutative unital ring]] is termed a '''local ring''' if it satisfies the following equivalent conditions:

			* It has a unique [[maximal ideal]]
			* There is exactly one homomorphism from the commutative unital ring whose image is a [[field]]
			* There is exactly one closed point in the [[spectrum]] (corresponding to the unique maximal ideal)

	===Definition with symbols===		===Definition with symbols===
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	* [[Field]]		* [[Field]]
	* [[Regular local ring]]		* [[Regular local ring]]
			* [[Completely primary ring]]

	===Weaker properties===		===Weaker properties===

Vipul at 09:13, 7 August 2007

2007-08-07T09:13:48Z

← Older revision		Revision as of 09:13, 7 August 2007
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	{{~~commring~~ property}}		{{curing property}}

	==Definition for commutative rings==		==Definition for commutative rings==

Vipul at 15:49, 30 June 2007

2007-06-30T15:49:39Z

← Older revision		Revision as of 15:49, 30 June 2007
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			{{commring property}}

	==Definition for commutative rings==		==Definition for commutative rings==

Vipul: /* Symbol-free definition */

2007-01-09T05:18:55Z

Symbol-free definition

New page

==Definition for commutative rings==

===Symbol-free definition===

A [[commutative unital ring]] is termed a '''local ring''' if it has a unique [[maximal ideal]].

===Definition with symbols===

{{fillin}}

==Definition for noncommutative rings==

{{further|[[Local ring (noncommutative rings)]]}}
==Relation with other properties==

===Stronger properties===

* [[Field]]
* [[Regular local ring]]

===Weaker properties===

* [[Semilocal ring]]
* [[Quasilocal ring]]
===Conjunction with other properties===

* [[Local Noetherian ring]]
* [[Local Artinian ring]]
* [[Local domain]]

[[Category: Properties of commutative rings]]