Study of Generalized Derivations in Rings with Involution

Let R be a prime ring with involution of the second kind and centre Z(R). Suppose R admits a generalized derivation F : R -> R associated with a derivation d : R -> R. The purpose of this paper is to study the commutativity of a prime ring R satisfying any one of the following identities: (i) F(x) circle x* is an element of Z(R) (ii) F([x, x*]) +/- x circle x* is an element of Z(R) (iii) F(x circle x*) +/- [x, x*] is an element of Z(R) (iv) F(x) circle d(x*) +/- x circle x* is an element of Z(R) (v) [F(x), d(x*)] +/- x circle x* is an element of Z(R) (vi) F(x) +/- x circle x* is an element of Z(R) (viii) F(x) +/- [x, x*] is an element of Z(R) (viii) [F(x), x*] -/+ F(x) circle x* is an element of Z(R) (ix) F(x circle x*) is an element of Z(R) for all x is an element of R.