Sigma-cotorsion rings

If CR is a flat Σ-cotorsion module, then CR admits a decomposition into a direct sum of indecomposable modules with a local endomorphism ring. The Jacobson radical J(S) of the endomorphism ring S=EndRC is characterized as the maximum ideal that acts locally T-nilpotently on CR. If R is semilocal and C=C(R), then the radical consists of those endomorphisms f:C→C whose image is contained in CJ.