Let $A,B,C$ be commutative rings with given ring homomorphisms $f:A\rightarrow C $ and $g: B \rightarrow C$. Let $A\times_C B:=\{(a,b)\in A \times B : f(a)=g(b)\}$ be their pullback (with the subring structure induced from $A \times B$).
Is there a good description of the prime ideals or maximal ideals of $A \times_C B$ in terms of the ideals of $A,B$ and $C$?
Thank you.
Motivated by this question: Prime ideals and maximal ideals of the Pullback of rings
If $A \to C$ is surjective, then we have $$\mathrm{Spec}(A \times_C B) = \mathrm{Spec}(A) \sqcup_{\mathrm{Spec}(C)} \mathrm{Spec}(B)$$ in the category of topological spaces, even in the category of ringed spaces (as well as in the category of schemes). This gives an explicit description of the points. In fact, $\mathrm{Spec}(B) \hookrightarrow \mathrm{Spec}(A \times_C B)$ is a closed immersion, and the complement is isomorphic to the complement of $\mathrm{Spec}(C) \hookrightarrow \mathrm{Spec}(A)$, so that as sets we have (which I denote by $|-|$) $$|\mathrm{Spec}(A \times_C B)| \cong |\mathrm{Spec}(B)| \sqcup |\mathrm{Spec}(A) \setminus \mathrm{Spec}(C)|.$$
Proofs, details and examples:
If $A \to C$ and $B \to C$ are not surjective, I don't think that we can say much. For example, let $A \subseteq C$ be a subring, $f$ be the inclusion, then $A \times_C B \cong \{b \in B : g(b) \in A\}$, and I doubt that we can say much about its prime ideals - it is just a subring of $B$, and every subring of $B$ has this form (consider $g = \mathrm{id}_B$).