Representations of Relatively Complemented Modular Lattices

Introduction. A module over a ring will be said to be locally projective if and only if every finitely generated submodule is projective. As will be shown (7.14), it readily follows from known facts that if M is a locally projective module over a regular ring R, then the set L(M, R) of all finitely generated submodules of M is a relatively complemented modular lattice. This paper is concerned with the representation problem suggested by the above observation. The fundamental theorem, 8.2, gives sufficient conditions in order for a relatively complemented modular lattice B to be isomorphic to L(M, R) for some locally projective module Mover a regular ring R. Essential use will be made of the results in Jónsson [7], and henceforth that paper will be referred to briefly as CM. In particular, the embedding theorem CM3.2 plays a fundamental role in the construction of the representation module. In CM this result was proved for complemented Arguesian lattices only, but an easy extension to the relatively complemented case is given in §1 of the present paper. §§2-8 are devoted to the construction of the representation module, and to the proof of the fundamental representation theorem. The method is roughly comparable to an approach to the classical coordinatization theorem for projective geometries that combines ideas found in Baer [2] and in Artin [l ] as follows: First the given space is embedded as a hyperplane in a space of higher dimension. The old space is regarded as the hyperplane at infinity and the new points as the points of an affine space. The translations of this space form an Abelian group P, and the trace-preserving endomorphisms of T form a division ring E. The group P is regarded as a vector space over E, and to each point at infinity there corresponds in an obvious manner a one-dimensional subspace of P. This yields the homogeneous coordinates of the points of the original space. To a remarkable extent this can be imitated here, although there are complications due to the fact that the given lattice has been embedded in a larger one, and it is therefore necessary to single out a subgroup To of "admis-