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Filial rings on direct sums and direct products of torsion-free abelian groups
E. I. Kompantsevaab, T. K. T. Nguyenc, V. A. Gazaryanbd
a 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.
Received: 20.12.2020
Accepted: 21.02.2021
Citation:
E. I. Kompantseva, T. K. T. Nguyen, V. A. Gazaryan, “Filial rings on direct sums and direct products of torsion-free abelian groups”, Chebyshevskii Sb., 22:1 (2021), 200–212
Linking options:
https://www.mathnet.ru/eng/cheb997 https://www.mathnet.ru/eng/cheb/v22/i1/p200
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