Let be rings, integral over ; let be prime ideals of such that and say. Then .
Question 1. Why ?
My attempt: Since , we have . But m is maximal in , which is not necessarily maximal in . I can't get by this.
Question 2. When we use that notation , which means the localization of at the prime ideal of . But in this corollary, doesn't necessarily be a prime ideal of . Why can he write ? Should we write rigorously?