Utumi Abelian groups

Abstract

In a recent paper written by Y. Ibrahim and M. Yousif (Comm. in Algebra, 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 ∼= B and A∩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 ⊕ L is a direct summand of M . In this paper, all the Utumi 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. Mathematics Subject Classification (2020).Primary: 20K21; Secondary: 20K30, 16D10.