I want to prove the following theorem, I have some idea and need some push:
Let $f \in \bar{k}[x,y]$ be a non-constant polynomial. Let $I$ be any non-zero ideal in $C_f = \bar{k}[x,y]/(f)$. Show that the prime ideals of $C_f$ that contain $I$ is a finite set.
My idea is that if I can show that $C_f /I$ has finitely many prime ideals, then we are done. This is because "prime ideals of $C_f$ containing $I$" are in 1-1 correspondence with the "ideals of prime ideals $C_f/I$". I could not done a nice setup, so any help would be great.