Krull's Hauptidealsatz (principal ideal theorem) says that for a Noetherian ring $R$ and any $r\in R$ which is not a unit or zero-divisor, all primes minimal over $(r)$ are of height 1. How badly can this fail if $R$ is a non-Noetherian ring? For example, if $R$ is non-Noetherian, is it possible for there to be a minimal prime over $(r)$ of infinite height?
EDIT: The answer is yes. See http://mathoverflow.net/questions/42510/how-badly-can-krulls-hauptidealsatz-fail-for-non-noetherian-rings