Associative rings whose adjoint semigroup is locally nilpotent

摘要

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 ${rcirc s}={r+s+rs}$ 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. Tasić. It is also shown that the adjoint group of a radical ring R is locally nilpotent if and only if R is locally Lie-nilpotent.