On generalized derivations and commutativity of prime and semiprime rings

Abstract

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 $in$ R. Let S be a nonempty 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:$(i) F(x)F(y) - xy in Z(R), (ii) F(x)F(y) + xy in Z(R), (iii) F{x)F(y) - yx in Z(R), (iv) F(x)F(y) + yx in Z{R), (v) F[x,y - [x,y] in Z(R), (vi) F[x,y] + [x,y] in Z(R), (vii) F(x o y) - x o y in Z(R)$, and (viii) $F(x o y) + x o y in Z(R)$, for all $x,y in S$.