In this paper, we describe ss-supplement submodules in terms of a special class of endomorphisms. Let R be a ring with semisimple radical and P be a projective R−module. We show that there is a bijection between ss-supplement submodules of P and ss-supplement submodules of EndR(P). Moreover, we define radical-s-projective modules as a generalization of projective modules. We prove that every ss-supplement submodule of a projective R−module is radical-s-projective over the ring R with semisimple radical. We show that over SSI -ring R, every radical-s-projective R−module is projective. We provide that over a ring R with semisimple radical, every ss-supplement submodule of a projective R−module is a direct summand if and only if every radical-s-projective R−module is projective.