Maximal ideal: Difference between revisions
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{{basicdef}} | |||
{{curing-ideal property}} | {{curing-ideal property}} | ||
{{quotient is a|field}} | {{quotient is a|field}} | ||
==Definition== | ==Definition== | ||
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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=== | |||
* [[Stronger than::Prime ideal]]: {{proofofstrictimplicationat|[[Maximal implies prime]]|[[prime not implies maximal]]}} | |||
* [[Stronger than::Radical ideal]] | |||
* [[Stronger than::Intersection of maximal ideals]] | |||
==Metaproperties== | |||
{{not intersection-closed ideal property}} | |||
{{not intringcondn ideal}} | |||
==Effect of property operators== | |||
{{applyingoperatorgives|intersection-closure|intersection of maximal ideals}} | |||
==External links== | |||
===Definition links=== | |||
* {{wp|Maximal ideal}} | |||
* {{planetmath|MaximalIdeal}} | |||
* {{mathworld|MaximalIdeal}} | |||
Latest revision as of 20:22, 17 January 2009
This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra
This article defines a property of an ideal in a commutative unital ring |View other properties of ideals in commutative unital rings
This property of an ideal in a ring is equivalent to the property of the quotient ring being a/an: field | View other quotient-determined properties of ideals in commutative unital rings
Definition
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 on makes into a simple -module.
Definition with symbols
An ideal in a commutative ring is termed maximal if it satisfies the following equivalent conditions:
- For any ideal such that ≤ ≤ , is equal either to or to .
- The quotient ring is a field.
Relation with other properties
Weaker properties
- Prime ideal: For proof of the implication, refer Maximal implies prime and for proof of its strictness (i.e. the reverse implication being false) refer prime not implies maximal
- Radical ideal
- Intersection of maximal ideals
Metaproperties
Intersection-closedness
This property of ideals in commutative unital rings is not closed under taking arbitrary intersections; in other words, an arbitrary intersection of ideals with this property need not have this property
Template:Not intringcondn ideal
Effect of property operators
The intersection-closure
Applying the intersection-closure to this property gives: intersection of maximal ideals