Rings on Abelian Torsion-Free Groups of Finite Rank

2021/05/22 by Kompantseva, Ekaterina, Tuganbaev, Askar · 1 citation
#16B99 #20K30 #20K99 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2105.10674

Abstract

In the class of reduced Abelian torsion-free groups G of finite rank, we describe TI-groups, this means that every associative ring on G is filial. If every associative multiplication on G is the zero multiplication, then G is called a nil_a-group. It is proved that a reduced Abelian torsion-free group G of finite rank is a TI-group if and only if G is a homogeneous Murley group or G is a nil_a-group. We also study the interrelations between the class of homogeneous Murley groups and the class of nil_a-groups. For any type t≠ (∞,∞,…) and every integer n>1, there exist 2^ℵ₀ pairwise non-quasi-isomorphic homogeneous Murley groups of type t and rank n which are nil_a-groups. We describe types t such that there exists a homogeneous Murley group of type t which is not a nil_a-group. This paper will be published in Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry.

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• Rings on quotient divisible abelian groups

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