On a group whose generalized torsion elements are torsion elements

We show that a group whose generalized torsion elements are torsion elements (which we call a TR*-group) is a torsion-by-R* group, an extension of a torsion group by a group without generalized torsion elements. We also discuss a generalized torsion group, a group all of whose non-trivial elements are generalized torsion elements.