Absolute ideals of mixed abelian groups

A ring on an abelian group $G$ is a ring, whose additive group is isomorphic to $G$. A subgroup $A$ of an abelian group $G$ is called its absolute ideal, if $A$ is an ideal in every ring on $G$. In 1973. L.Fuchs formulated the problem of describing abelian groups, on which there exists a ring structure, whose every ideal is absolute. Such abelian group is call a $RAI$-group. A group $G$ is a group of class $K$, if its $p$-component $T_p(G)$ is a separable and unbounded group for all prime $p$ such that $T_p(G) \ne 0$ and every multiplication on the torsion subgroup $T(G)$ can be uniquely continued to a multiplication on $G$. In this work, a description of countable $RAI$-groups of class $K$ is given.