Intersection of maximal ideals: Difference between revisions
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An [[ideal]] in a [[commutative unital ring]] is termed an '''intersection of maximal ideals''' if it can be expressed as an intersection of [[maximal ideal]]s (this is really a tautological definition). | An [[ideal]] in a [[commutative unital ring]] is termed an '''intersection of maximal ideals''' if it can be expressed as an intersection of [[maximal ideal]]s (this is really a tautological definition). | ||
Note that although a maximal ideal is assumed to be proper, an ''intersection of maximal ideals'' could be the whole ring, i.e. we allow the empty intersection. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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A ring in which every radical ideal is an intersection of maximal ideals is termed a [[Jacobson ring]] (or sometimes a Hilbert ring). | A ring in which every radical ideal is an intersection of maximal ideals is termed a [[Jacobson ring]] (or sometimes a Hilbert ring). | ||
==Metaproperties== | |||
{{intersection-closed ideal property}} | |||
Revision as of 15:25, 24 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: semisimple ring | View other quotient-determined properties of ideals in commutative unital rings
Definition
Symbol-free definition
An ideal in a commutative unital ring is termed an intersection of maximal ideals if it can be expressed as an intersection of maximal ideals (this is really a tautological definition).
Note that although a maximal ideal is assumed to be proper, an intersection of maximal ideals could be the whole ring, i.e. we allow the empty intersection.
Relation with other properties
Stronger properties
Weaker properties
- Radical ideal: For full proof, refer: Intersection of maximal ideals implies radical
Related ring properties
A ring in which every radical ideal is an intersection of maximal ideals is termed a Jacobson ring (or sometimes a Hilbert ring).
Metaproperties
Intersection-closedness
This property of ideals in commutative unital rings is intersection-closed: an arbitrary intersection of ideals with this property, also has this property