Radicals of Rings with Quotient Divisible Additive Groups

In this paper, we study rings on groups from the class QD1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{Q}\mathcal{D}1$$\end{document} of quotient divisible abelian groups of rank 1. The Jacobson radical and the upper nil-radical of rings on groups from QD1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{Q}\mathcal{D}1$$\end{document} are described. This description allowed in QD1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{Q}\mathcal{D}1$$\end{document} 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 is an element of QD1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\in \mathcal{Q}\mathcal{D}1$$\end{document} such that the rings on G are determined by their upper nil-radicals. For groups from the class QD1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{Q}\mathcal{D}1$$\end{document}, 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 QD1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{Q}\mathcal{D}1$$\end{document}. It is also shown that for any group G is an element of QD1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\in \mathcal{Q}\mathcal{D}1$$\end{document}, 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.