Irreducible ideal
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