Indecomposable flat cotorsion modules

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 U R 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 R R 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.