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

Let R be a prime ring of characteristic different from 2, C be its extended centroid and Qr be its right Martindale quotient ring and f(t1, ..., tn) 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)) = 0 for all s = (s1, ..., sn) ∈ Rn. Then either F = 0 or d = 0, except when d is an inner derivation of R, there exists λ ∈ C such that F(r) = λr for all r ∈ R and f(t1, ..., tn)2 is central valued on R.