Irreducible ideal: Difference between revisions
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{{curing-ideal property}} | |||
{{quotient is a|irreducible ring}} | |||
==Definition for commutative rings== | ==Definition for commutative rings== | ||
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* [[Primary ideal]] under the assumption that the ring is [[Noetherian ring|Noetherian]] {{proofat|[[Irreducible implies primary (Noetherian)]]}} | * [[Primary ideal]] under the assumption that the ring is [[Noetherian ring|Noetherian]] {{proofat|[[Irreducible implies primary (Noetherian)]]}} | ||
[[ | ===Incomparable properties=== | ||
[[ | |||
* [[Primary ideal]] (for non-Noetherian rings) | |||
* [[Radical ideal]] | |||
Latest revision as of 16:24, 12 May 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: irreducible ring | View other quotient-determined properties of ideals in commutative unital rings
Definition for commutative rings
Symbol-free definition
An ideal in a commutative unital ring is termed irreducible if it satisfies the following equivalent conditions:
- It cannot be expressed as an intersection of two ideals properly containing it
- The quotient ring by that ideal is an irreducible ring
Definition for noncommutative rings
The symbol-free definition carries over verbatim from the commutative case.
Relation with other properties
Stronger properties
Weaker properties
- Primary ideal under the assumption that the ring is Noetherian For full proof, refer: Irreducible implies primary (Noetherian)
Incomparable properties
- Primary ideal (for non-Noetherian rings)
- Radical ideal