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

• Maximal ideal
• Prime ideal

Weaker properties

• Primary ideal under the assumption that the ring is Noetherian For full proof, refer: Irreducible implies primary (Noetherian)