On Rings Whose Quasi-Projective Modules Are Projective or Semisimple

For two modules \(M\) and \(N\), \(P_M(N)\) stands for the largest submodule of \(N\) relative to which \(M\) is projective. For any module \(M\), \(P_M(N)\) defines a left exact preradical. It is given some properties of \(P_M(N)\).\ We express \(P_M(N)\) as a trace submodule. In this paper, we study rings with no quasi-projective modules other than semisimples and projectives, that is, rings whose quasi-projectives are either projective or semisimple (namely QPS-ring). Semi-Artinian rings and rings with no right p-middle class are characterized by using this functor: a ring \(R\) right semi-Artinian if and only if for any right \(R\)-module \(M\), \(P_M(M)\leq_e M\).