Maximal ideal - Revision history

Vipul: /* Relation with other properties */

2009-01-17T20:22:44Z

Relation with other properties

← Older revision		Revision as of 20:22, 17 January 2009
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	===Weaker properties===		===Weaker properties===

	* [[Prime ideal]]: {{proofofstrictimplicationat\|[[Maximal implies prime]]\|[[prime not implies maximal]]}}		* [[Stronger than::Prime ideal]]: {{proofofstrictimplicationat\|[[Maximal implies prime]]\|[[prime not implies maximal]]}}
	* [[Radical ideal]]		* [[Stronger than::Radical ideal]]
	* [[Intersection of maximal ideals]]		* [[Stronger than::Intersection of maximal ideals]]

	==Metaproperties==		==Metaproperties==

Vipul: 7 revisions

2008-05-12T16:27:07Z

7 revisions

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

Vipul at 01:01, 19 February 2008

2008-02-19T01:01:57Z

← Older revision		Revision as of 01:01, 19 February 2008
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			{{basicdef}}
	{{curing-ideal property}}		{{curing-ideal property}}

	{{quotient is a\|field}}		{{quotient is a\|field}}

	~~{{basicdef}}~~
	==Definition==		==Definition==

Vipul: /* Weaker properties */

2008-01-24T14:51:56Z

Weaker properties

← Older revision		Revision as of 14:51, 24 January 2008
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	===Weaker properties===		===Weaker properties===

	* [[Prime ideal]]		* [[Prime ideal]]: {{proofofstrictimplicationat\|[[Maximal implies prime]]\|[[prime not implies maximal]]}}
	* [[Radical ideal]]		* [[Radical ideal]]
	* [[Intersection of maximal ideals]]		* [[Intersection of maximal ideals]]

Vipul: /* Metaproperties */

2008-01-10T01:28:03Z

Metaproperties

← Older revision		Revision as of 01:28, 10 January 2008
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	{{not intersection-closed ideal property}}		{{not intersection-closed ideal property}}

	{{intringcondn ~~ideal}}~~		{{not intringcondn ideal}}

	~~{{transfercondn~~ ideal}}

	==Effect of property operators==		==Effect of property operators==

Vipul at 22:17, 5 January 2008

2008-01-05T22:17:58Z

← Older revision		Revision as of 22:17, 5 January 2008
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	* For any ideal <math>J</math> such that <math>M</math> ≤ <math>J</math> ≤ <math>R</math>, <math>J</math> is equal either to <math>M</math> or to <math>R</math>.		* For any ideal <math>J</math> such that <math>M</math> ≤ <math>J</math> ≤ <math>R</math>, <math>J</math> is equal either to <math>M</math> or to <math>R</math>.
	* The quotient ring <math>R/M</math> is a [[field]].		* The quotient ring <math>R/M</math> is a [[field]].

			==Relation with other properties==

			===Weaker properties===

			* [[Prime ideal]]
			* [[Radical ideal]]
			* [[Intersection of maximal ideals]]

			==Metaproperties==

			{{not intersection-closed ideal property}}

			{{intringcondn ideal}}

			{{transfercondn ideal}}

			==Effect of property operators==

			{{applyingoperatorgives\|intersection-closure\|intersection of maximal ideals}}

			==External links==

			===Definition links===

			* {{wp\|Maximal ideal}}
			* {{planetmath\|MaximalIdeal}}
			* {{mathworld\|MaximalIdeal}}

Vipul at 20:11, 5 January 2008

2008-01-05T20:11:20Z

← Older revision		Revision as of 20:11, 5 January 2008
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	{{quotient is a\|field}}		{{quotient is a\|field}}

			{{basicdef}}
	==Definition==		==Definition==

Vipul at 09:35, 7 August 2007

2007-08-07T09:35:21Z

@@ Line 1: / Line 1: @@
-==Definition for commutative rings==
+{{curing-ideal property}}
 ===Symbol-free definition===
@@ Line 15: / Line 19: @@
 * For any ideal <math>J</math> such that <math>M</math> &le; <math>J</math> &le; <math>R</math>, <math>J</math> is equal either to <math>M</math> or to <math>R</math>.
 * The quotient ring <math>R/M</math> is a [[field]].

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

2007-01-09T07:41:12Z

Definition for non-commutative rings

New page

==Definition for commutative rings==

===Symbol-free definition===

An [[ideal]] in a [[commutative unital ring]] (or more generally, in any [[commutative ring]]) is termed '''maximal''' if it is proper (not the whole ring) and it satisfies the following equivalent conditions:

* There is no ideal of the ring properly in between this ideal and the whole ring
* The quotient of the ring by this ideal is a [[field]]
* The natural action of <math>R</math> on <math>R/M</math> makes <math>R/M</math> into a simple <math>R</math>-module.

===Definition with symbols===

An ideal <math>M</math> in a commutative ring <math>R</math> is termed '''maximal''' if it satisfies the following equivalent conditions:

* For any ideal <math>J</math> such that <math>M</math> ≤ <math>J</math> ≤ <math>R</math>, <math>J</math> is equal either to <math>M</math> or to <math>R</math>.
* The quotient ring <math>R/M</math> is a [[field]].

==Definition for non-commutative rings==

For non-commutative rings, there are three notions:

* [[Maximal two-sided ideal]]: Maximal among [[two-sided ideal]]s
* [[Maximal left ideal]]: Maximal among [[left ideal]]s
* [[Maximal right ideal]]: Maximal among [[right ideal]]s

[[Category: Properties of ideals in commutative rings]]
[[Category: Quotient-determined properties of ideals in commutative rings]]