Commutativity results for periodic rings (Q1187320)

The following theorem is proved: Let \(n\) be a positive integer and let \(R\) be an \(n(n+1)\)-torsion free periodic ring with \(N(R)\) commutative. If \(R\) satisfies one of the following conditions: (*) \((xy)^ n-x^ ny^ n\in Z(R)\) and \((xy)^{n+1}- x^{n+1}y^{n+1}\in Z(R)\) for every \(x,y\in R\); (**) \((xy)^ n-x^ ny^ n\in Z(R)\) and \((xy)^{n+1}- y^{n+1}x^{n+1}\in Z(R)\) for every \(x,y\in R\); then \(R\) is commutative (\(N(R)\) denotes the set of nilpotent elements of \(R\)). This theorem generalizes some results obtained by I. N. Herstein, V. Gupta, A. Yaqub and the author.