Irreducible ideal

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.

Stronger properties

 * Maximal ideal
 * Prime ideal

Weaker properties

 * Primary ideal under the assumption that the ring is Noetherian

Incomparable properties

 * Primary ideal (for non-Noetherian rings)
 * Radical ideal