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 Mprojective if and only if for N ≤ T ≤ M , whenever T /N 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 GVring. In addition, we prove that if every δ-cosingular right R-module is semisimple, then Z(M) is a direct summand of M for every right R-module M if and only if Z δ (M) is a direct summand of M for every right R-module M .

Rings for which every cosingular module is discrete

“…We denote by σ[M ] the full subcategory of R-modules whose objects are all R-modules subgenerated by M (see [18]). In [15], the present authors work on rings for which every (simple) cosingular module is projective. They show that for a ring R, every simple cosingular R-module is projective if and only if R is a GV (GCO) ring.…”

“…It is known by [9,Theorem 2.3] that a ring R is right perfect if and only if every quasi-projective R-module is discrete. Inspired by [6] and [15], in this paper, we study rings R (resp., modules M ) such that every (resp., M -)cosingular R-module (resp., in σ[M ]) is discrete. We call them CD-rings (resp., CD-modules).…”

“…In [15], Talebi In [14], the present authors work on rings for which every (simple) cosingular module is projective. They show that for a ring R, every simple cosingular R-module is projective if and only if R is a GV (GCO) ring.…”