Associative rings whose adjoint semigroup is locally nilpotent

The set of all elements of an associative ring R. not necessarily with a unit element, forms a semigroup R-ad under the circle operation r o s = r + s + rs on R. The ring R is called radical if R-ad is a group. It is proved that the semigroup R-ad is nilpotent of class n (in sense of A. Mal'cev or B. H. Neumann and T. Taylor) if and only if the ring R is Lie-nilpotent of class n. This yields a positive answer to a question posed by A. Krasil'nikov and independently considered by D. Riley and V. Tasic. It is also shown that the adjoint group of a radical ring R is locally nilpotent if and only if R is locally Lienilpotent.