here's a doubt which arised from a previous question:
Suppose $R$ is a ring and $S \subseteq R$ is a subring. Moreover suppose $R$ is integral over $S$ and $R$ is finitely generated as an $S$-module. In a previous post Matt E showed that we always have Specmax(S) is bounded above by Specmax(R). My question is: is it true that for any maximal ideal of S there are at most t maximal ideals of R?