On associative rings with locally nilpotent adjoint semigroup

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 r circle s = r + s + rs for all r, s in R. This semigroup is locally nilpotent if every finitely generated subsemigroup of R-ad is nilpotent (in sense of A. I. Mal'cev or B. H. Neumann and T. Taylor). The ring R is locally Lie-nilpotent if every finitely generated subring of R is Lie-nilpotent. It is proved that R-ad is a locally nilpotent semigroup if and only if R is a locally Lie-nilpotent ring. [References: 7]