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Krull's height theorem says that if $R$ is a Noetherian ring and $I$ is a proper ideal generated by $n$ elements of $R$, then $\operatorname{ht} I\le n$.

When $R$ is not Noetherian, this is not true. I wonder if there exist other conditions, (instead of Noetherian), that give the result? Is there other inequalities in non-Noetherian case?

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One (partial) answer to the question:
Anderson, Dobbs, Eakin, and Heinzer, in the paper "On the generalized principal ideal theorem and krull domains" have:

"If $R \subset T$ is an integral extension of domains and $R$ is Noetherian, then $T$ satisfies (the conclusion of the) generalized principal ideal theorem"

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