TITLE: Representation theorem for modular lattices?
QUESTION [18 upvotes]: Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices?
For example, I wonder: Does every modular lattice embed into the lattice of submodules of some module? After all, I cannot think of any relation which holds in the lattice of submodules of some module, except for modularity.

REPLY [7 votes]: By adding a couple of conditions we can indeed obtain a representation theorem as the OP suggests.  Specifically, von Neumann's coordinatization theorem says that every complemented modular lattice of order at least 4 is isomorphic to the lattice of principal right ideals of a von Neumann regular ring.  See the wikipedia article on continuous geometry or any of the books on continuous geometry mentioned in this mathoverflow post.
If we replace modularity with the stronger Arguesian law mentioned in Todd and Pedro's answers, we get the more general coordinatization theorem of Jónsson, which says that every complemented Arguesian lattice with a large partial 3-frame is again isomorphic to the lattice of principal right ideals of a von Neumann regular ring.  See Proposition 10.1 of this paper by Wehrung for a first order characterization of "having a large partial 3-frame".