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 , and hence, it must be irreducible.