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

Examples

Properties

An element of an integral domain is prime if and only if the ideal generated by it is a non-zero prime ideal. This is not true for irreducible ideals; an irreducible ideal may be generated by an element that is not an irreducible element, as is the case in \mathbb Z for the ideal 4 \mathbb Z since it is not the intersection of two strictly greater ideals. In algebraic geometry, if an ideal I of a ring R is irreducible, then V(I) is an irreducible subset in the Zariski topology on the spectrum. The converse does not hold; for example the ideal in defines the irreducible variety consisting of just the origin, but it is not an irreducible ideal as.