Let $S \subseteq R$ be commutative rings with 1 and suppose $Spec(R)$ is a Noetherian topological space. How do we show that the number of each $T \in Spec(R)$ lying over $P \in Spec(S)$ is finite?
I guess the idea is to use the going up theorem. We let $T \in Spec(R)$ such that $T$ lies over $P$. I don't see how to relate $T$ with a closed set of the form $V(J)$ for some ideal $J$ and then how to produce a descending chain of closed subsets of $Spec(R)$ so that we can use the Noetherian condition to guarantee this chain stabilize?