ON GENERALIZED DERIVATIONS AND COMMUTATIVITY OF PRIME AND SEMIPRIME RINGS

Let R be a prime ring and θ, Φ endomorphisms of R. An addi-tive mapping F : R right arraw R is called a generalized (θ, Φ )-derivationon R if there exists a (θ, Φ )-derivation d : R right arraw R such thatF(xy) = F(x)θ(y) + (x)d(y) for all x, y Φ R. Let S be a non-empty subset of R. In the present paper for various choices of Swe study the commutativity of a semiprime (prime) ring R admit-ting a generalized (θ,Φ )-derivation F satisfying any one of the prop-erties: (i) F(x)F(y) − xy element of Z(R), (ii) F(x)F(y) + xy element of Z(R),(iii) F(x)F(y) − yx element ofZ(R), (iv) F(x)F(y) + yx element of Z(R),(v) F[x, y] − [x, y] element of Z(R), (vi) F[x, y] + [x, y] element of Z(R),(vii) F(x ◦ y) − x ◦ y element of Z(R), and (viii) F(x ◦ y) + x ◦ y element of Z(R),for all x, y element of S.