A right [Formula: see text]-module [Formula: see text] is called a dual-Utumi-module (DU-module) if for any two proper submodules [Formula: see text] and [Formula: see text] of [Formula: see text] with [Formula: see text] and [Formula: see text], there exist summands [Formula: see text] and [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text] with [Formula: see text], [Formula: see text] small submodules of [Formula: see text] and [Formula: see text] a direct summand of [Formula: see text]. In this work, we characterize rings [Formula: see text] for which every (finitely generated, cyclic, free) [Formula: see text]-module is DU-module.

No abstract

In a recent paper written by Y. Ibrahim and M. Yousif (2018), the following class of modules is considered: a right R -module M is called a Utumi module if, whenever A and B are submodules of M with A\cong B and A\cap B=0 , there exist direct summands K and L of M such that A is essential in K , B is essential in L and K\oplus L is a direct summand of M . In this paper, all the Utumi \mathbb{Z} -modules (i.e. Abelian groups) and some special classes of these are determined. As an application, it is proved that all the pseudo-continuous Abelian groups are quasi-continuous.