Rings on quotient divisible abelian groups

In this paper, we consider the class [Formula: see text] of mixed abelian quotient groups of torsion free rank 1. In the class [Formula: see text], we describe groups [Formula: see text] such that there exists a ring on [Formula: see text] in which every ideal is an absolute ideal (a fully invariant subgroup) of [Formula: see text]. Thus, Problem 93 of L. Fuchs’s monograph “Infinite Abelian Groups, Vol. II, New York-London: Academic Press, 1973” is solved in the class [Formula: see text]. In addition, in the class [Formula: see text], we describe groups in which every absolute ideal is a fully invariant subgroup.