Levitzki Radical for certain Varieties

Abstract

Let A be a nonassociative algebra. We let An denote the subalgebra generated by all products of n elements from A. Inductively we define A(0) = A and A(n+1) = (A(n))2. We say that A is nilpotent if, for some n, An = {0}. A is solvable if A(n) = {0} for some n. An algebra is locally nilpotent (locally solvable) if each finitely generated subalgebra is nilpotent (solvable). In this paper will always be some variety of algebras defined by a set of homogeneous identities. We say that local nilpotence is a radical property in if each contains a maximal locally nilpotent ideal L and A/L has no non-zero locally nilpotent ideals. The ideal L is then called the Levitzki radical of A.