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2 I wanted to inform people about the fact that it came from somewhere else, but then I thought it might be read as an attempt to garner votes there too

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?

(disclosure: this is being promoted from http://math.stackexchange.com/questions/6994/how-badly-can-krulls-hauptidealsatz-fail-for-non-noetherian-rings)

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# How badly can Krull's Hauptidealsatz fail for non-Noetherian rings?

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?

(disclosure: this is being promoted from http://math.stackexchange.com/questions/6994/how-badly-can-krulls-hauptidealsatz-fail-for-non-noetherian-rings)