ON n-COMMUTING AND n-SKEW-COMMUTING MAPS WITH GENERALIZED DERIVATIONS IN PRIME AND SEMIPRIME RINGS

Let R be a ring with center Z(R), let n be a fixed positive integer, and let I be a nonzero ideal of R. A mapping h : R -> R is called n-centralizing (n-commuting) on a subset S of R if [h(x), x(n)] is an element of Z(R) ([h(x), x(n)] = 0 respectively) for all x is an element of S. The following are proved: (1) if there exist generalized derivations F and G on an n!-torsion free semiprime ring R such that F(2) + G is n-commuting on R, then R contains a nonzero central ideal; (2) if there exist generalized derivations F and G on an n!-torsion free prime ring R such that F(2) + G is n-skew-commuting on I, then R is commutative.