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.