irreducible variety

A pair of crossing lines is really two separate pieces glued at a point; a single smooth circle is one indivisible piece. Irreducibility makes this intuition precise: an irreducible variety is one that cannot be honestly broken into two smaller closed parts. It is the geometric notion of being 'a single thing'.

Formally, a nonempty topological space (or variety) is irreducible if it is not the union of two proper closed subsets, equivalently if every nonempty open subset is dense, equivalently if any two nonempty open sets intersect. For an affine variety X this is a property of its algebra: X is irreducible if and only if its ideal I(X) is prime, equivalently its coordinate ring k[X] is an integral domain.

Every variety decomposes essentially uniquely. In a Noetherian space — and varieties are Noetherian — any closed set is a finite union of irreducible closed sets, and after discarding redundant ones this list of irreducible components is unique. So the irreducible varieties are the indivisible atoms, and a general variety is glued from finitely many of them; this is the geometric counterpart of the primary decomposition of ideals.