Associative rings with metabelian adjoint group

The set of all elements of an associative ring R, not necessarily with a unit element, forms a monoid under the circle operation r s=r+s+rs on R whose group of all invertible elements is called the adjoint group of R and denoted by R°. The ring R is radical if R=R°. It is proved that a radical ring R is Lie metabelian if and only if its adjoint group R° is metabelian. This yields a positive answer to a question raised by S. Jennings and repeated later by A. Krasilʹnikov. Furthermore, for a ring R with unity whose multiplicative group R* is metabelian, it is shown that R is Lie metabelian, provided that R is generated by R* and R modulo its Jacobson radical is commutative and artinian. This implies that a local ring is Lie metabelian if and only if its multiplicative group is metabelian.