Some conditions for the commutativity of rings (Q1332969)

This is the latest in a long series of papers, by various authors, on rings satisfying identities of the form \([x,y^ n] = [x^ n, y]\) or related conditions. The authors establish two results: (I) If \(n > 1\) and \(R\) is a semiprime ring satisfying the identity \([x, [x^ n,y] - [x,y^ n]] = 0\), then \(R\) is commutative; (II) If \(m > 1\) and \(n \geq 1\) and \(R\) is a ring with 1 satisfying the identity \([x, x^ n y - xy^ m] = 0\), then \(R\) is commutative.