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?



How badly can Krull's Hauptidealsatz fail for non-Noetherian rings?
I think that the answer is **yes**.


Indeed, there are examples of integral domains $D$ such that every non-zero prime ideal of $D$ has infinite height.


Look at the paper 


"Anti-archimedean rings and power series rings"


D.D. Anderson; B.G. Kang; M H. Park


Communications in Algebra, 1532-4125, Volume 26, Issue 10, 1998, Pages 3223 – 3238.


