Some generalizations in certain classes of rings with involution

Let R be a 2-torsion free sigma-prime ring with an involution sigma, I a nonzero sigma-ideal of R. In this paper we explore the commutativity of R satisfying any one of the properties: (i) d(x) circle F(y) = 0 for all x, y is an element of I. (ii) [d(x), F(y)] = 0 for all x, y is an element of I. (iii) d(x) circle F(y) = x circle y for all x, y is an element of I. (iv) d(x)F(y) - xy is an element of Z(R) for all x, y is an element of I. We also discuss (alpha, beta)-derivations of sigma-prime rings and prove that if G is an (alpha, beta)-derivation which acts as a homomorphism or as an anti-homomorphism on I, then G = 0 or G = beta on I.