Commutativity of rings satisfying some polynomial conditions (Q1911351)

Let \(R\) denote a ring, \(Z\) the center, \(N\) the set of nilpotent elements, and \(J\) the Jacobson radical; and let \(n\) be a fixed positive integer. It is shown that if \(R\) has 1 and is \(n\)-torsion free, then \(R\) must be commutative if the following two conditions are satisfied: (i) \(x^n y^n = y^n x^n\) for all \(x, y \in R\setminus J\) (resp. \(R \setminus N\)); (ii) \((xy)^{n + 1} - x^{n + 1} y^{n + 1} \in Z\) for all \(x, y \in R \setminus J\) (resp. \(R \setminus N\)). The other theorem asserts that if \(R\) is periodic and \(n(n + 1)\)-torsion-free, and if \(N\) is commutative and \((xy)^n - y^n x^n \in Z\) for all \(x, y \in R \setminus N\), then \(R\) is commutative.