Definition:Induced Mapping on Spectra of Rings

Definition
Let $A$ and $B$ be commutative rings with unity.

Let $\operatorname{Spec} A$ and $\operatorname{Spec} B$ be their spectra.

Let $f : A \to B$ be a ring homomorphism.

The induced mapping on spectra by $f$ is the mapping $f^* : \operatorname{Spec} B \to \operatorname{Spec} A$ with:
 * $f^* (\mathfrak p) = f^{-1}(\mathfrak p)$,

the preimage of a prime ideal $\mathfrak p \in \operatorname{Spec} B$.

Also denoted as
The induced map on spectra by $f$ is also denoted $\operatorname{Spec} f$; see the spectrum functor.

Also see

 * Preimage of Prime Ideal under Ring Homomorphism is Prime Ideal
 * Induced Mapping on Prime Spectra is Continuous
 * Definition:Spectrum of Ring Functor