δss-supplemented modules and rings

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 RSoc(RR) {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.