\begin{definition}[Definition:Induced Mapping on Maximal Spectra]
Let $k$ be a field.
Let $A$ and $B$ be finitely generated $k$-algebras.
Let $f : A \to B$ be a $k$-algebra homomorphism.
The '''induced mapping on spectra by $f$''' is the mapping $f^* : \operatorname{Max} B \to \operatorname{Max} A$ between their maximal spectra with:
:$f^* (\mathfrak m) = f^{-1}(\mathfrak m)$,
the preimage of a maximal ideal $\mathfrak m \in \operatorname{Max} B$.
\end{definition}