There's a well-known theorem for distributive lattices commonly referred to as the "prime ideal theorem:"
Let $L$ be a distributive lattice, $I$ an ideal of $L$, and $F$ a filter of $L$ such that $I\cap F=\varnothing$. Then there exists a prime ideal $P$ such that $I\subseteq P$ and $P\cap F=\varnothing$.
Is there an analogous theorem for modular lattices? A cursory look through one or two texts and a brief online search seem to indicate that the answer is no, but perhaps I am mistaken.
Does anyone have a counter-example or a reference to an affirmative answer? Thanks!
Take the modular lattice $M_3=\{0,a,b,c,1\}$ where $0,1$ are the minimum and maximum elements, respectively, and consider $I=\{0,c\}$ and $F=\{a,1\}$. There is no (proper) prime ideal containing $I$, and hence none disjoint from $F$. E.g., $I$ is not prime since $a\wedge b=0\in I$ but $a,b\notin I$.