# Prime ideals and maximal ideals of the Pullback of rings

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 $$C$$ ?

• it is the pushout of the prime spectra of $A$ and $B$ along the prime spectrum of $C$. – quantum Jun 10 at 21:37
• @quantum: push out of the corresponding spectrum in which category ? certainly it can't be in the category of sets ... – user102248 Jun 10 at 23:10
• Not category of sets but of topological spaces. Anyway the underlying set would be this (I believe). It shouldn't be too far from and the rough reason is because the images and preimages of a ring from a (commutative unitary) ring homomorphism are all rings. – quantum Jun 12 at 6:32

$$\newcommand\spec{\mathrm{Spec}}$$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.