On Rings whose Quasi Projective Modules Are Projective or Semisimple

Abstract

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) <=(e) M.