Artinian rings with supersolvable 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 a ° b = a + b + ab. The group of all invertible elements of this monoid is called the adjoint group of R and is denoted by R ^°. It is proved that an artinian ring R with supersolvable adjoint group R ^° must be Lie supersolvable. An example of a Lie supersolvable ring with non-supersolvable adjoint group is also constructed. Received: 7 December 2007