Separable torsion free Abelian groups with $UA$-rings of endomorphisms

A semigroup $(R,\cdot)$ is said to be a unique addition ring ($UA$-ring) if there exists a unique binary operation $+$, making $(R,\cdot,+)$ into a ring. We call an abelian group $\operatorname{End}$-$UA$-group if its endomorphism ring is $UA$-ring. As a result we have obtained a characterization of separable tortion free $\operatorname{End}$-$UA$-groups.