Commutativity of rings with conditions on commutators, nilpotent and potent elements (Q1112932)

Let N denote the set of nilpotents, C the center of the ring R (not necessarily with a unit), \(n>1\) an integer, \(E_ n=\{x\in R|\) \(x^ n=x\}\). The main result is as follows: If (i) \(R=N+E_ n\); (ii) For any \(x,y\in R\), there exist \(f(\lambda),g(\lambda)\in Z[\lambda]\) such that \([x-x^ 2f(x),y-y^ 2g(y)]=0\); (iii) For all \(x,y\in R\), \((xy)^ n- (yx)^ n\in C\) then R is commutative. (iii) can be replaced by other conditions, in particular if, additionally, R is normal \((E_ 2\subset C)\) or has a unit. [Cf. \textit{H. Tominaga}, Math. Jap. 33, No.5, 809-811 (1988; see the preceding review).]