Maximal ideal: Difference between revisions
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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}} | |||
Revision as of 22:17, 5 January 2008
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
This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra
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
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
Intermediate subring condition
This property of ideals satisfies the intermediate subring condition for ideals: if an ideal has this property in the whole ring, it also has this property in any intermediate subring
View other properties of ideals satisfying the intermediate subring condition
Transfer condition
This property of ideals satisfies the transfer condition for ideals: if an ideal satisfies the property in the ring, its intersection with any subring satisfies the property inside that subring
Effect of property operators
The intersection-closure
Applying the intersection-closure to this property gives: intersection of maximal ideals