Rings on Abelian $\mathrm{MT}$-groups

Rings on $\mathrm{MT}$-groups are investigated, a mixed Abelian group $G$ is called an $\mathrm{MT}$-group if every multiplication on the torsion part of $G$ uniquely extends to a multiplication on $G$. The absolute Jacobson radical and the absolute nil-radical of $\mathrm{MT}$-groups are described (Problem 94 in the book Infinite Abelian Groups by L. Fuchs).