Commutativity of *-Prime Rings with Generalized Derivations

Abstract

Let R be a 2-torsion free ∗-prime ring and F be a generalized derivation of R with associated derivation d. If U is a ∗-Lie ideal of R then in the present paper, we shall show that U ⊆ Z(R) if R admits a generalized derivation F (with associated derivation d)  satisfying any one of the properties: (i) F[u,v]=[F(u),v],  (ii) F(u_o_v)=F(u)o_v_, (iii) F[u,v]=[F(u),v]+[d(v),u],  (iv) F(u_o_v)=F(u)o_v_+ d(v)o_u_, (v) F(uv)±_uv_=0 and(vi) d(u)F(v)±_uv_=0 for all u,v ∈ U.