Let $A,B,C$ be commutative Noetherian rings with given surjective ring homomorphisms $f:A\twoheadrightarrow C $ and $g: B \twoheadrightarrow C$. Let $A\times_C B:=\{(a,b)\in A \times B : f(a)=g(b)\}$ ( with the subring structure induced from $A \times B$ ) be the Pullback (https://en.wikipedia.org/wiki/Pullback_(category_theory) ).
Is there a good description of the prime ideals or maximal ideals of $A\times_C B$ in terms of ideals of $A,B$ and $C$ ?
$\newcommand\spec{\mathrm{Spec}}$Hint: You may be inspired by pure product, which is a special type of fiber product when $C=0$ (the zero ring). In this case the prime ideals are all the elements in the product of $\spec A\cup \{A\}$ and $\spec B \cup \{B\}$ without $A\times B$.
Since your homomorphism $f:A\rightarrow C$ and $g:B\rightarrow C$ are surjective it is easier to describe this (this is a general construction and we did not need $A$ and $B$ Noetherian), since the projections $A\times_C B \rightarrow A$ and $A\times_C B \rightarrow B$ of prime ideals in $A\times_C B$ are either prime ideals or the whole ring.
Things become more messier if $f$ or $g$ were not surjective.