Modules With Epimorphisms Between Their Submodules

An R ‐module M is called weakly uniserial if its submodules are comparable regarding embedding, i.e., if for any two submodules N , K of M , Hom R ( N , K ) or Hom R ( K , N ) contains an injective element. Here, we are interested in studying modules which for any two submodules of them there is an epimorphism from one to the other. Such a module is said to have the epicly related submodules property (we will say it has the ERSP, in short). In this paper, in addition to providing the properties of modules that have the ERSP, we show that every projective module over a principal right ideal domain has the ERSP. Also, every projective module over a local right hereditary ring has the ERSP. Then we prove that a ring R is an Artinian simple ring if and only if every right R ‐module has the ERSP. Among applications of our results, we classify quasi‐injective abelian group and finitely generated abelian groups that have the ERSP.