Irreducible ideal

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 ${\displaystyle \mathbb{Z}}$ for the ideal ${\displaystyle 4\mathbb{Z}}$ since it is not the intersection of two strictly greater ideals.

In algebraic geometry, if an ideal ${\displaystyle I}$ of a ring ${\displaystyle R}$ is irreducible, then ${\displaystyle V(I)}$ is an irreducible subset in the Zariski topology on the spectrum ${\displaystyle \mathrm{Spec}R}$. The converse does not hold; for example the ideal ${\displaystyle (x^2,xy,y^2)}$ in ${\displaystyle \mathbb{C}[x,y]}$ defines the irreducible variety consisting of just the origin, but it is not an irreducible ideal as ${\displaystyle (x^2,xy,y^2)=(x^2,y)\cap (x,y^2)}$.