Sigma-cotorsion rings

Abstract

It is proved that a ring R is right perfect if and only if it is Σ-cotorsion as a right module over itself. Several other conditions are shown to be equivalent. For example, that every pure submodule of a free right R-module is strongly pure-essential in a direct summand, or that the countable direct sum C(R) (ℵ 0) of the cotorsion envelope of R R is cotorsion. If C R is a flat Σ-cotorsion module, then C R 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=End R C is characterized as the maximum ideal that acts locally T-nilpotently on C R . If R is semilocal and C= C( R), then the radical consists of those endomorphisms f : C→C whose image is contained in CJ.