Indecomposable flat cotorsion modules

Abstract

An additive functor from the category of flat right R-modules to the category of abelian groups is continuous if it is isomorphic to a functor of the form−⊗R M, where M is a left R-module. It is shown that for any simple subfunctor A of−⊗ M there is a unique indecomposable flat cotorsion module UR for which A(U)≠0. It is also proved that every subfunctor of a continuous functor contains a simple subfunctor. This implies that every flat right R-module may be purely embedded into a product of indecomposable flat cotorsion modules. If CE(R) is the cotorsion envelope of RR and S= End;R CE(R), then a local ring monomorphism is constructed from R/J(R) to S/J(S). This local morphism of rings is used to associate a semiperfect ring to any semilocal ring. It also proved that if R is a semilocal ring and M a simple left R-module, then the functor−⊗R M on the category of flat right R-modules is uniform, and therefore contains a unique simple subfunctor.