Rings with (x, y, z) – (z, y, x) in the right nucleus

Abstract

Let R be a semiprime nonassociative ring satisfying (x, y, z)–(z, y, x) ∈ Nr then Nl = Nr where Nl and Nr are Lie ideals of R, the set {x ∈ Nr : (R, R, R)x = 0} = {x ∈ Nl : x(R, R, R) = 0} is an ideal of R, and it is contained in the nucleus. Further if [R, R]Nr ⊂ Nr and R is a prime ring with Nr ≠ 0 then R is either associative or commutative.