ON GENERALIZED DERIVATIONS AND COMMUTATIVITY OF PRIME AND SEMIPRIME RINGS

Let R be a prime ring and theta, phi endomorphisms of R. An additive mapping F : R -> R is called a generalized (theta, phi)-derivation on R if there exists a (theta, phi)-derivation d : R -> R such that F(xy) = F(x)theta(y) + phi(x)d(y) for all x, y is an element of R. Let S be a non-empty subset of R. In the present paper for various choices of S we study the commutativity of a semiprime (prime) ring R admitting a generalized (theta, phi)-derivation F satisfying any one of the properties: (i) F(x) F(y) - xy is an element of Z(R), (ii) F(x)F(y) + xy is an element of Z(R), (iii) F(x) F(y) - yx is an element of Z(R), (iv) F(x)F(y) + yx is an element of Z(R), (v) F[x, y] - [x, y] is an element of Z(R), (vi) F[x, y] + [x, y] is an element of Z(R), (vii) F(x circle y) - x circle y is an element of Z(R), and (viii) F(x circle y) + x circle y is an element of Z(R), for all x, y is an element of S.