As is well-known, the lattice of submodules of a module is modular. What I did not know is that the converse is not true, and that lattices of submodules must satisfy a stronger property, the arguesian law.
The Arguesian law is a lattice-theoretic analogue of Desargues’ theorem in projective geometry. I read the statement of the theorem several times and I have no intuition about what it means.
There is a kind of converse to this result: a complemented lattice can be embedded into the lattice of submodules of a module if and only if it is arguesian. (I found the result in Gratzer’s book on lattice theory, which is viewable in Google Books.)