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 to[Formula: see text] which are pairwise non-isomorphic.