Irreducible ideal

In mathematics, an ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two larger ideals.

Every prime ideal is irreducible. Every irreducible ideal of a Noetherian ring is a primary ideal, and consequently for Noetherian rings an irreducible decomposition is a primary decomposition. Every primary ideal of a principal ideal domain is an irreducible ideal. Every irreducible ideal is a primal ideal.