Let R —> S be a ring map. Let p ⊂ R be a prime. Assume that

1. there exists a unique prime q ⊂ S lying over p, and
2. either
   1. going up holds for R —> S, or
   2. going down holds for R —> S and there is at most one prime of S above every prime of R.

Then S_p=S_q. See Tag 00EA.