Abstract
Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d (r omicron s)(r omicron s) + (r omicron s) d (r omicron s)(n) - d (r omicron s))(m) for all r, s epsilon I, then R is commutative. (ii) If (d (r omicron s)(r omicron s) + (r omicron s) d (r omicron s)(n) - d (r omicron s)(m) epsilon Z(R) for all r, s epsilon I, then R satisfies s(4), the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.