Coweakly Uniserial Modules and Rings Whose (2-Generated) Modules Are Coweakly Uniserial

A module is called weakly uniserial if for any two its submodules at least one of them is embedded in the other. This is a nontrivial generalization of uniserial modules and rings. Here, we introduce and study the dual of this concept. In fact, an R-module M is called coweakly uniserial if for any submodules N,K of M,HomRM/N,M/K or HomRM/K,M/N contains a surjective element. In this paper, in addition to presenting the properties of this concept, we show that a ring R is homogeneous semisimple if and only if every (projective) right R-module is coweakly uniserial. Also, in a semi-Artinian ring R, it is shown that if every 2-generated right R-module is coweakly uniserial, then R is a homogeneous semisimple ring. Then, we prove that Q has no coweakly uniserial subgroups. Among applications of our results, we classify quasi-continuous, quasi-injective, and uniform abelian groups that are coweakly uniserial.