On prime and semiprime rings with derivations

Let R be a ring and S a nonempty subset of R. A mapping f : R --> R is called commuting on S if [f (x), x] = 0 for all x is an element of S. In this paper, firstly, we generalize the well-known result of Posner related to commuting derivations on prime rings. Secondly, we show that if R is a semiprime ring and I is a nonzero ideal of R, then a derivation d of R is commuting on I if one of the following conditions holds: (i) For all x, y is an element of I, either d([x, y]) = [x, y] or d([x, y]) = - [x, y]. (ii) For all x, y is an element of I, either d(x circle y) = x circle y or d(x circle y) = - (x circle y). (iii) R is 2-torsion free, and for all x, Y is an element of I, either [d(x), d(y)] = d([x, y]) or [d(x), d(y)] = d([y, x]). Furthermore, if d(I) not equal {0}, then R has a nonzero central ideal. Finally, we introduce the notation of generalized biderivation and prove that every generalized biderivation. on a noncommutative prime ring is a biderivation.