Radical Rings with Soluble Adjoint Groups

An associative ring R, not necessarily with an identity, is called radical if it coincides with its Jacobson radical, which means that the set of all elements of R forms a group denoted by R^∘ under the circle operation r ∘ s = r + s + rs on R. It is proved that every radical ring R whose adjoint group R^∘ is soluble must be Lie-soluble. Moreover, if the commutator factor group of R^∘ has finite torsion-free rank, then R is locally nilpotent.