Herstein’s theorem for generalized derivations in rings with involution

Let $R$ be an associative ring. An additive map $F:R\toR$ is called a generalized derivation if there exists a derivation $d$ of $R$ such that $Finsert ignore into journalissuearticles values(xy);=Finsert ignore into journalissuearticles values(x);y+xdinsert ignore into journalissuearticles values(y);$ for all $x,y\in R$. In [7], Herstein proved the following result: If $R$ is a prime ring of $charinsert ignore into journalissuearticles values(R);\neq 2$ admitting a nonzero derivation $d$ such that $[dinsert ignore into journalissuearticles values(x);,dinsert ignore into journalissuearticles values(y);]=0$ for all $x,y\in R$, then $R$ is commutative. In the present paper, we shall study the above mentioned result for generalized derivations in rings with involution.