Commutativity of rings with constraints on nilpotents and nonnilpotents (Q1123966)

Let R be a ring, N its set of nilpotent elements, and \(n>1\) a fixed positive integer. It is proved that R is commutative if it satisfies the following conditions: (i) N is commutative; (ii) \(x^ ny=xy^ n\) for all \(x,y\in R\setminus N\); (iii) if \(a\in N\), \(b\in R\) and \(n![a,b]=0\), then \([a,b]=0\). None of the three conditions can be deleted.