Commutativity Results for Rings with Potent K-Engel Elements

Abstract

Let R be a ring. First we show that if : (P) For each a, b ∈R  there exist positive integers n =  n(a, b)  >  1, m  =  m(a, b)  ≥ 1 and k = k(a, b) ≥ 1, such that [amb, a]n = [amb, a]k, and Nr(R) = 0, then R is commutative. Furthermore, we prove that any s-unital ring satisfying(P)must have a nil commutator ideal. Finally we show that an s-unital ring R is commutative if: (P ′) For each a,b ∈ R there exists positive integer n = n(a,b) > 1 such that (b−br)([xmy, a]−[amb, a]n) = 0, where r > 1 and m ≥ 1 are fixed positive integers, and (Ir) the commutator ideal of R is r!-torsion free.