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, HomR(M/N, M/K) or HomR(M/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.