# 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 $f^* : \operatorname{Spec} B \to \operatorname{Spec} A$ between their spectra 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.