Spectrum of direct product is disjoint union of spectra

This article gives the statement, and possibly proof, of a fact about how a property of a homomorphism of commutative unital rings, forces a property for the induced map on spectra
View other facts about induced maps on spectra

Statement

Verbal statement

The spectrum of a direct product of commutative unital rings is the disjoint union of their spectra (both as a set, and as a topological space).

Category-theoretic statement

The spectrum, viewed as a contravariant functor, from the category of commutative unital rings to the category of sets (resp. the category of topological spaces or the category of locally ringed spaces) converts products to coproducts.