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 $\mathfrak p \in \Spec B$.

Induced morphism of locally ringed spaces

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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