δss-supplemented modules and rings

In this paper, we introduce the concept of δ_ss-supplemented modules and provide the various properties of these modules. In particular, we prove that a ring R is δ_ss-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 δ_ss-supplemented. We define projective δ_ss-covers and prove the rings with the property that every (simple) module has a projective δ_ss-cover are δ_ss-supplemented. We also study on δ_ss-supplement submodules.