Rings on quotient divisible abelian groups

Abstract

The paper is devoted to the study of absolute ideals of groups in the class $\mathcal{QD}1$, which consists of all quotient divisible abelian groups of torsion-free rank 1. A ring is called an $AI$-ring (respectively, an $RF$-ring) if it has no ideals except absolute ideals (respectively, fully invariant subgroups) of its additive group. An abelian group is called an $RAI$-group (respectively, an $RFI$-group) if there exists at least one $AI$-ring (respectively, $FI$-ring) on it. If every absolute ideal of an abelian group is a fully invariant subgroup, then this group is called an $afi$-group. It is shown that every group in $\mathcal{QD}1$ is an $RAI$-group, an $RFI$-group, and an $afi$-group. Thus, Problem 93 of L. Fuchs' monograph \emph{``Infinite Abelian Groups, Vol. II, New York-London: Academic Press, 1973''} is resolved within the class $\mathcal{QD}1$. For any group in $\mathcal{QD}1$, all rings on it that are $AI$-rings are described. Furthermore, the set of all $AI$-rings on $G \in \mathcal{QD}1$ coincides with the set of all $FI$-rings on $G$. In addition, the principal absolute ideals of groups in $\mathcal{QD}1$ are described.