δss -supplemented modules and rings

Abstract

Abstract In this paper, we introduce the concept of δss -supplemented modules and provide the various properties of these modules. In particular, we prove that a ring R is δss -supplemented as a left module if and only if R S o c ( R R ) {R \over {Soc\left( {_RR} \right)}} is semisimple and idempotents lift to Soc( RR) if and only if every left R-module is δss -supplemented. We define projective δss -covers and prove the rings with the property that every (simple) module has a projective δss -cover are δss -supplemented. We also study on δss -supplement submodules.