Definition:Spectrum of Ring Functor

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Let $\mathbf{Ring}$ be the category of commutative rings with unity.

To topological spaces

Let $\mathbf{Top}$ be the category of topological spaces.

The spectrum functor $\Spec {} : \mathbf{Ring} \to \mathbf{Top}$ is the contravariant functor with:

Object functor:         $\Spec A$ is the spectrum of a ring $A$
Morphism functor: If $f : A \to B$ is a ring homomorphism, $\Spec f : \Spec B \to \Spec A$ is the induced map on spectra