SOME COMMUTATIVITY THEOREMS CONCERNING ADDITIVE MAPPINGS AND DERIVATIONS ON SEMIPRIME RINGS

Let R be a ring with its center Z(R) and I a nonzero ideal of R. The purpose of this paper is to investigate identities satisfied by additive mappings on prime and semiprime rings. More precisely, we prove the following result. Let R be a semiprime ring, and let F, d : R -> R be two additive mappings such that F(xy) = F(x)y + xd(y) for all x, y is an element of R. If F(xy) +/- xy is an element of Z(R) for all x, y is an element of I, then [d(x), x] = 0 for all x is an element of I. Further, if d is a derivation such that d(I) not equal (0), then R contains a nonzero central ideal. Moreover, if R is prime and d is a derivation such that d(I) not equal (0), then R is commutative.