Rings on almost completely decomposable Abelian groups

The absolute radical of an Abelian group $G$ is the intersection of radicals of all associative rings with additive group $G$. L. Fuchs formulated the problem on a description of absolute radicals of Abelian groups. For a group from some class of almost completely decomposable Abelian groups the absolute Jacobson radical is described. In the class of almost completely decomposable Abelian groups semisimple groups are described.