DERIVATIONS OF PRIME AND SEMIPRIME RINGS

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and n a fixed positive integer. (i) If (d(x)y+xd(y)+d(y)x+yd(x))(n) = xy+yx for all x, y is an element of I, then R is commutative. (ii) If char R not equal 2 and (d(x)y + xd(y) + d(y)x + yd(x))(n) - (xy + yx) is central for all x, y is an element of I, then R is commutative. We also examine the case where R is a semiprime ring.