Let $D$ be a domain, $\mathfrak{a,b,p}\subsetneq D$ ideals with $\mathfrak{p}$ maximal and $\mathfrak{a}$ invertible (there is some $\mathfrak{c}$ ideal with $\mathfrak{ac}=\mu D$, with $\mu\in D\setminus\{0\}$).
If $\mathfrak{ap\subseteq b\subseteq a}$ then $\mathfrak{b=ap \;\vee \; b=a}$.
I've been working for a hour on this but don't get nothing right. Please some hint or help.
Maybe I'm missing something, but $\mathfrak p\subseteq\mathfrak b\mathfrak a^{-1}\subseteq D$ gives immediately what you want.