Spectrum of integral domain is irreducible

This article gives a fact about the relation between ring-theoretic assumptions about a commutative unital ring and topological consequences for the spectrum
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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.