Generators for complemented modular lattices and the von Neumann-Jnsson coordinatization theorems

Extending work of von Neumann, Jnsson has shown that each complemented modular lattice, L admitting a large partial n-frame with n a parts per thousand yen 4, or with n a parts per thousand yen 3 and L Arguesian, can be coordinatized as the lattice of all principal right ideals of some regular ring. His proof built on the embedding of L into the subgroup lattice of an abelian group which follows from Frink's embedding of L into to a direct product of subspace lattices of irreducible projective spaces and coordinatization of the latter. We offer a proof which, in addition to these results, employs only some elementary linear algebra. Luca Giudici's thesis [6] is an important source for this approach.