The Classification of Torsion-free TI-Groups

An abelian group Formula: see text is called a Formula: see text-group if every associative ring with the additive group Formula: see text is filial. The filiality of a ring Formula: see text means that the ring Formula: see text is associative and all ideals of any ideal of Formula: see text are ideals in Formula: see text. In this paper, torsion-free Formula: see text-groups are described up to the structure of associative nil groups. It is also proved that, for torsion-free abelian groups that are not associative nil, the condition Formula: see text implies the indecomposability and homogeneity. The paper contains constructions of Formula: see text such groups of any rank from 2 toFormula: see text which are pairwise non-isomorphic.