Differential Identities and Generalized Derivations in Prime Rings with Involution

摘要

Let R be a *-ring with the center Z(R). In the present paper we study commutativity of a prime ring which admits derivations d, g satisfying any one of the properties: (i) d([x, x*]) ± [x, x*] ? Z(R), (ii) d(x ? x*) ± x ? x* ? Z(R), (iii) d([x, x*]) ± x ? x* ? Z(R), (iv) d(x ? x*) ± [x, x*] ? Z(R), (v) d(x) ? d(x*) ± x ? x* ? Z(R), (vi) d(x)g(x*) ± [x, x*] ? Z(R) and (vii) d(x)g(x*) ± x ? x* ? Z(R), for all x ? R. Further, some related results are also obtained if R admits a generalized derivation F satisfying either of the properties F(xx*) ± F(x)F(x*) ? Z(R) for all x ? R or F(xx*) ± F(x*)F(x) ? Z(R) for all x ? R. Finally, an example has been provided to justify the hypotheses in various results.