Filial rings on direct sums and direct products of torsion-free abelian groups

E. I. Kompantseva^ab, T. K. T. Nguyen^c, V. A. Gazaryan<sup>bd

a</sup> Moscow Pedagogical State University (Moscow)
^b Financial University under the Government of the Russian Federation (Moscow)
^c Vietnam education cooperation joint stock company (Vietnam)
^d Moscow State University named after M.V. Lomonosov (Moscow)

Abstract: A ring whose additive group coincides with an abelian group $G$ is called a ring on $G$. An abelian group $G$ is called a $TI$-group if every associative ring on $G$ is filial. If every (associative) ring on an abelian group $G$ is an $SI$-ring (a hamiltonian ring), then $G$ is called an $SI$-group (an $SI_H$-group). In this article, $TI$-groups, $SI_H$-groups and $SI$-groups are described in the following classes of abelian groups: almost completely decomposable groups, separable torsion-free groups and non-measurable vector groups. Moreover, a complete description of non-reduced $TI$-groups, $SI_H$-groups and $SI$-groups is given. This allows us to only consider reduced groups when studying $TI$-groups.

Keywords: abelian group, ring on a group, filial ring, $TI$-group.

UDC: 512.541

Received: 20.12.2020
Accepted: 21.02.2021

DOI: 10.22405/2226-8383-2018-22-1-200-212