TI-Groups in the Class of Quotient Divisible Abelian Groups

We consider the class QD1 of mixed abelian quotient divisible groups of torsion-free rank 1. Groups from this class, as well as countable mixed abelian groups of rank 1 are determined by their own torsion part and the class of equivalent height-matrices, which are invariants of such groups. On the other hand, every group from QD1 can be described by its cocharacteristic. An abelian group G is called a $$TI$$ -group if every associative ring on G is filial. In the class QD1 $$TI$$ -groups are described in the language of these invariants.