Rings For Which Every Cosingular Module Is Projective

Abstract

Let R be a ring and M be an R-module. In this paper we investigate modules M such that every (simple) cosingular R-module is M-projective. We prove that every simple cosingular module is M-projective if and only if for N <= T <= M, whenever TAN is simple cosingular, then N is a direct summand of T. We show that every simple cosingular right R-module is projective if and only if R is a right GV-ring. It is also shown that for a right perfect ring R, every cosingular right R-module is projective if and only if R is a right GV-ring. In addition, we prove that if every delta-cosingular right R-module is semisimple, then (Z) over bar (M) is a direct summand of M for every right R-module M if and only if (Z) over bar (delta)(M) is a direct summand of M for every right R-module M.