Intersection of maximal ideals: Difference between revisions
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===Weaker properties=== | ===Weaker properties=== | ||
* [[Radical ideal]] | * [[Radical ideal]]: {{proofat|[[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). | |||
Revision as of 01:11, 10 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).
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).