Karl W. Gruenberg

Two theorems on Engel groups

Let G be a group and let us write [ x , y ] = x −1 y −1 xy . We shall say that G is an Engel group , or that it satisfies the Engel condition , if to any pair of elements x , y in G there can be assigned an integer k (which is allowed to depend on x , y ) such that

Commutators in associative rings

Let R be an arbitrary associative ring, and X a set of generators of R . The elements of X generate a Lie ring, [ X ], say, with respect to the addition and subtraction in R , and the multiplication [ a, b ] = ab − ba . In this note we shall be concerned with the following question: if [ X ] is given to be nilpotent as a Lie ring, what does this imply about R ?