FREE MODULES WITH 2 DISTINGUISHED SUBMODULES

Over a commutative ring R with identity, free modules M with 2 disting uished submodules are studied. The category Rep(2)R of such objects M have the obvious morphisms between them, which are homomorphisms betwe en R-modules preserving the distinguished submodules. The endomorphism s for each M constitute a subalgebra End(R)M of the algebra End(R)M, a nd the realizability of lambda-generated R-algebras A as End(R)M is co nsidered for cardinals lambda. Despite the fact that 4 is the minimal number of distinguished submodules for realizing any algebra over a fi eld R, we are able to prove a similar result in Rep(2)R for many rings R including R = Z and algebras which are cotorsion-free. Several exam ples illustrate the boarder line of our main result. The main theorem is applied for constructing Butler groups in [11].