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Spectrum of a ring
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== Functorial perspective == It is useful to use the language of [[category theory]] and observe that <math>\operatorname{Spec}</math> is a [[functor]]. Every [[ring homomorphism]] <math>f: R \to S</math> induces a [[continuous function (topology)|continuous]] map <math>\operatorname{Spec}(f): \operatorname{Spec}(S) \to \operatorname{Spec}(R)</math> (since the preimage of any prime ideal in <math>S</math> is a prime ideal in <math>R</math>). In this way, <math>\operatorname{Spec}</math> can be seen as a contravariant functor from the [[category of commutative rings]] to the [[category of topological spaces]]. Moreover, for every prime <math>\mathfrak{p}</math> the homomorphism <math>f</math> descends to homomorphisms :<math>\mathcal{O}_{f^{-1}(\mathfrak{p})} \to \mathcal{O}_\mathfrak{p} </math> of local rings. Thus <math>\operatorname{Spec}</math> even defines a contravariant functor from the category of commutative rings to the category of [[locally ringed space]]s. In fact it is the universal such functor, and hence can be used to define the functor <math>\operatorname{Spec}</math> up to [[natural isomorphism]].{{citation needed|date=August 2015}} The functor <math>\operatorname{Spec}</math> yields a contravariant [[equivalence of categories|equivalence]] between the [[category of commutative rings]] and the '''category of affine schemes'''; each of these categories is often thought of as the [[opposite category]] of the other.
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