TITLE: In a finite ring extension there are only finitely many prime ideals lying over a given prime ideal
QUESTION [10 upvotes]: I'm trying to solve the exercise 6.7 of Miles Reid's Undergraduate Commutative Algebra (pag 93). 

How can I prove that if $B$ is a finite ring extension of $A$, there are only finitely many prime ideals of $B$ whose intersection with $A$ is a given prime ideal of $A$? 

Thank you!

REPLY [2 votes]: Hint: First show that it's enough to prove this for maximal $\mathfrak p$ (localize).  Then show that it's enough to prove this when the maximal ideal is $(0)$ (factor). Then you have a finite $k$-algebra and must show that it has finitely many prime ideals.