Utumi Abelian groups

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.