Rings for which every cosingular module is discrete

Abstract

In this paper we introduce the concepts of CD-rings and CD-modules. Let R be a ring and M be an R-module. We call R a CD-ring in case every cosingular R-module is discrete, and M a CD-module if every M-cosingular R-module in σ[M ] is discrete. If R is a ring such that the class of cosingular R-modules is closed under factor modules, then it is proved that R is a CD-ring if and only if every cosingular R-module is semisimple. The relations of CD-rings are investigated with V-rings, GV-rings, SC-rings, and rings with all cosingular R-modules projective. If R is a semilocal ring, then it is shown that R is right CD if and only if R is left SC with Soc(R R) essential in R R. Also, being a V-ring and being a CD-ring coincide for local rings. Besides of these, we characterize CD-modules with finite hollow dimension.

EKFN-Modules

“…M is semilocal and finitely generated Lemma 6. (Proposition 2.26 of [9]) Let R be a commutative domain. Then the following are equivalent.…”