A note on b-generalized derivations with a quadratic equation in prime rings

Let R be a prime ring of characteristic different from ۲, C be its extended centroid and Q_{r} be its right Martindale quotient ring and f(t_{۱},...,t_{n}) be a  multilinear polynomial over C, which is not central valued on R. Assume that F is a b-generalized derivation on R and d is a derivation of R such that F(f(s))d(f(s))+d(f(s))F(f(s))=۰for all s=(s_{۱},...,s_{n})\in R^{n}. Then either F=۰ or d=۰, except when d is an inner derivation of R, there exists \lambda \in C such that F(r)=\lambda r for  all r\in R and f(t_{۱},...,t_{n})^{۲} is central valued on R.

Keywords:

b-generalized derivation , multilinear polynomial , prime ring , generalized polynomial identity