Principal ideal ring - Revision history

Vipul: /* Relation with other properties */

2009-02-08T23:24:29Z

Relation with other properties

← Older revision		Revision as of 23:24, 8 February 2009
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	* [[Stronger than::Bezout ring]]		* [[Stronger than::Bezout ring]]
	* [[Stronger than::Noetherian ring]]		* [[Stronger than::Noetherian ring]]
	* [[Stronger than::One-dimensional ring]]: {{proofat\|Principal ideal ring implies one-dimensional]]}}		* [[Stronger than::One-dimensional ring]]: {{proofat\|[[Principal ideal ring implies one-dimensional]]}}

	==Metaproperties==		==Metaproperties==

Vipul: /* Relation with other properties */

2009-02-08T23:24:14Z

Relation with other properties

← Older revision		Revision as of 23:24, 8 February 2009
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	* [[Stronger than::Bezout ring]]		* [[Stronger than::Bezout ring]]
	* [[Stronger than::Noetherian ring]]		* [[Stronger than::Noetherian ring]]
			* [[Stronger than::One-dimensional ring]]: {{proofat\|Principal ideal ring implies one-dimensional]]}}

	==Metaproperties==		==Metaproperties==

Vipul: /* Metaproperties */

2009-02-06T00:44:56Z

Metaproperties

← Older revision		Revision as of 00:44, 6 February 2009
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	{{localization-closed curing property}}		{{localization-closed curing property}}

	~~More generally, the~~ [[localization ~~at a multiplicatively~~ closed ~~subset~~]] ~~of a principal ideal ring, is still a principal ideal ring.~~		{{proofat\|[[Principal ideal ring is localization-closed]]}}

Vipul: /* Relation with other properties */

2009-01-17T16:53:58Z

Relation with other properties

← Older revision		Revision as of 16:53, 17 January 2009
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	===Conjunction with other properties===		===Conjunction with other properties===

	* [[Principal ideal domain]] is a principal ideal ring which is also an [[integral domain]]		* [[Weaker than::Principal ideal domain]] is a principal ideal ring which is also an [[integral domain]]

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

	* [[Bezout ring]]		* [[Stronger than::Bezout ring]]
	* [[Noetherian ring]]		* [[Stronger than::Noetherian ring]]

	==Metaproperties==		==Metaproperties==

Vipul: 5 revisions

2008-05-12T16:33:33Z

5 revisions

← Older revision	Revision as of 16:33, 12 May 2008
(No difference)

Vipul: /* Metaproperties */

2008-03-15T22:56:09Z

Metaproperties

← Older revision		Revision as of 22:56, 15 March 2008
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	{{localization-closed curing property}}		{{localization-closed curing property}}

			More generally, the [[localization at a multiplicatively closed subset]] of a principal ideal ring, is still a principal ideal ring.

Vipul: /* Metaproperties */

2008-03-15T22:55:20Z

Metaproperties

← Older revision		Revision as of 22:55, 15 March 2008
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	{{Q-closed curing property}}		{{Q-closed curing property}}

			{{localization-closed curing property}}

Vipul at 23:02, 7 January 2008

2008-01-07T23:02:01Z

← Older revision		Revision as of 23:02, 7 January 2008
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	==Metaproperties==		==Metaproperties==

	{{Q-closed}}		{{Q-closed curing property}}

Vipul at 23:01, 7 January 2008

2008-01-07T23:01:15Z

← Older revision		Revision as of 23:01, 7 January 2008
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			{{curing property}}
	==Definition==		==Definition==

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

	A [[commutative unital ring]] ~~(or any [[commutative ring]])~~ is termed a '''principal ideal ring''' if every [[ideal]] in it is [[principal ideal\|principal]], that is, if every ideal is generated by a single element.		A [[commutative unital ring]] is termed a '''principal ideal ring''' if every [[ideal]] in it is [[principal ideal\|principal]], that is, if every ideal is generated by a single element.

	===Definition with symbols===		===Definition with symbols===
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	* [[Noetherian ring]]		* [[Noetherian ring]]

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

			{{Q-closed}}

Vipul: Started the page

2007-01-09T05:00:04Z

Started the page

New page

==Definition==

===Symbol-free definition===

A [[commutative unital ring]] (or any [[commutative ring]]) is termed a '''principal ideal ring''' if every [[ideal]] in it is [[principal ideal|principal]], that is, if every ideal is generated by a single element.

===Definition with symbols===

{{fillin}}

==Relation with other properties==

===Conjunction with other properties===

* [[Principal ideal domain]] is a principal ideal ring which is also an [[integral domain]]

===Weaker properties===

* [[Bezout ring]]
* [[Noetherian ring]]

[[Category: Properties of commutative rings]]