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 $2^{{\aleph }_0}$ such groups of any rank from 2 to$2^{{\aleph }_0}$ which are pairwise non-isomorphic.