Krull's Principal Ideal Theorem in non-Noetherian settings

Let $P$ be a finitely generated ideal of a commutative ring $R$. Krull's Principal Ideal Theorem states that if $R$ is Noetherian and $P$ is minimal over a principal ideal of $R$, then $P$ has height at most one. Straightforward examples show that this assertion fails if $R$ is not Noetherian. We consider what can be asserted in the non-Noetherian case in place of Krull's theorem.