On central identities equipped with skew Lie product involving generalized derivations

Abstract

Let R be a *-ring. For any x,y∈R, we denote the skew Lie product of x and y by ▿[x,y]=xy-yx∗. An additive mapping F:R→R is called a generalized derivation if there exists a derivation d such that F(xy)=F(x)y+xd(y) for all x,y∈R. The objective of this paper is to chracterize generalized derivations and to describe the structure of prime rings with involution * involving skew Lie product. In particular, we prove that if R is a 2-torsion free prime ring with involution * of the second kind and admits a generalized derivation (F,d) such that ▿[x,F(x∗)]±▿[x,x∗]∈Z(R) for all x∈R, then R is commutative or F=∓IR, where IR is the identity mapping of R. Moreover, some related results are also obtained. Finally, we provide two examples to prove that the assumed restrictions on our main results are not superfluous.