 Irreducible ideal

A proper ideal $I$ is irreducible if $I=J\cap K$ implies $I=J$ or $I=K$. In a Noetherian ring, every irreducible ideal is a primary ideal; every proper ideal is a finite intersection of irreducible ideals.

In the context of ring theory, an **irreducible ideal** is a specific type of ideal in a ring that has certain properties.