Representations of relatively complemented modular lattices

Bjarni J{ónsson
1962 Transactions of the American Mathematical Society  
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
more » ... 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
doi:10.1090/s0002-9947-1962-0140457-8 fatcat:62up6d7ttjfbrg6xy5wspd7eo4