Prime ideals of a subring of $A\times B$

Question

Let $A$, $B$ and $C$ are commutative rings and $\alpha:A\longrightarrow C$ and $\beta:B\longrightarrow C$ homomorphisms of commutative rings. Let $$A\times_C B=\{(a,b)\in A\times B : \alpha (a)=\beta (b)\}.$$ Simply this is a subring of $A\times B$. We know the prime ideals of $A\times B$, that is, $p\times B$ or $A\times q$ where $p$ and $q$ are prime ideals of $A$ and $B$ respectively. But what about of prime ideals of $A\times_C B$? Thanks for helping me. (In my work $\beta$ is one to one.)