ON COMMUTATIVITY THEOREMS FOR P. I. - RINGS WITH UNITY - KEZLAN, THOMAS P. (Tamkang Journal of Mathematics, 1993)
The purpose of this paper is to show how a previous commutativity theorem for general rings can be used to prove commutativity theorems for rings with unity, and to obtain several new results via this route, e.g., if a ring with unity satisfies either x^k[x^n, y] = [x, y^m]x^ or x^k[x^n,y] = [x,y^m]y^(m > 1) and if either (A) m and n are relatively prime or (B) n[x,y]=0 implies [x,y]=0, then R is commutative.