The Classification of Torsion-free TI-Groups

An abelian group A is called a TI-group if every associative ring with the additive group A is filial. The filiality of a ring R means that the ring R is associative and all ideals of any ideal of R are ideals in R. In this paper, torsion-free TI-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 TI implies the indecomposability and homogeneity. The paper contains constructions of 2(N0) such groups of any rank from 2 to 2(N0) which are pairwise non-isomorphic.