Spectrum of integral domain is irreducible

Statement
The spectrum of an integral domain is an irreducible space: it cannot be expressed as a union of two proper closed subsets.

Converse
The converse is not in general true. For instance, for a local Artinian ring, the spectrum is just a single point, which is irreducible, but the ring is not in general an integral domain.

More generally, the spectrum of a ring is irreducible if and only if there is a unique minimal prime ideal, or equivalently, if the nilradical is a prime ideal.

Proof
In fact, the spectrum is the closure of a single point: the prime ideal $$0$$, and hence, it must be irreducible.