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.