In this paper, we study rings on groups from the class \(\mathcal{Q}\mathcal{D}1\) of quotient divisible abelian groups of rank 1. The Jacobson radical and the upper nil-radical of rings on groups from \(\mathcal{Q}\mathcal{D}1\) are described. This description allowed in \(\mathcal{Q}\mathcal{D}1\) to characterize the groups on which the rings are determined by their Jacobson radicals. It is shown that the class of such groups coincides with the class of groups \(G\in \mathcal{Q}\mathcal{D}1\) such that the rings on G are determined by their upper nil-radicals. For groups from the class \(\mathcal{Q}\mathcal{D}1\) , the absolute Jacobson radical and the absolute nil-radical are described. Thus, Problem 94 of the L. Fuchs’ monograph “Infinite Abelian Groups” [Vol. II. New York-London: Academic Press, 1973] is solved for groups in \(\mathcal{Q}\mathcal{D}1\) . It is also shown that for any group \(G\in \mathcal{Q}\mathcal{D}1\) , its absolute Jacobson radical and absolute nil-radical are realized as the Jacobson radical and upper nil-radical, respectively, of some associative and commutative ring on G.
