Rings on Abelian torsion-free groups of finite rank

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 types t and \(n\in \mathbb {N}\) and rank n which are \(nil_a\)-groups. We describe types t and \(n\in \mathbb {N}\) such that there exists a homogeneous Murley group of type t and rank n which is not a \(nil_a\)-group.