Definition:Induced Mapping on Spectra of Rings

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

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

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

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

Also denoted as
The induced map on spectra by $f$ is also denoted $\Spec f$; 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