Rings on Abelian Torsion-Free Groups of Finite Rank

TL;DR

This paper characterizes TI-groups within reduced Abelian torsion-free groups of finite rank, establishing their structure as either homogeneous Murley groups or $nil_a$-groups, and explores their interrelations and diversity.

Contribution

It provides a complete characterization of TI-groups in this class and constructs numerous examples of homogeneous Murley groups that are $nil_a$-groups, expanding understanding of their structure.

Findings

TI-groups are either homogeneous Murley groups or $nil_a$-groups.

Existence of many non-quasi-isomorphic homogeneous Murley groups of given types and ranks.

Identification of types where homogeneous Murley groups are not $nil_a$-groups.

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 $t\ne (\infty,\infty,\ldots)$ and every integer $n>1$, there exist $2^{\aleph_0}$ 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 Beitr\"{a}ge zur Algebra und Geometrie / Contributions to Algebra and Geometry.