Co-centralizing generalized derivations acting on multilinear polynomials in prime rings

Let R be a noncommutative prime ring ofý ýcharacteristic different from 2ý, ýU the Utumi quotient ring of Rý, ýC (=Z(U)) the extended centroidý ýof Rý. ýLet 0≠a∈R and f(x1,…,xn) a multilinearý ýpolynomial over C which is noncentral valued on Rý. ýSupposeý ýthat G and H are two nonzero generalized derivations of Rý ýsuch that a(H(f(x))f(x)−f(x)G(f(x)))∈C for allý ýx (x1,…,xn)∈Rný. ý one of the following holdsý: ýf(x1,…,xn)2 is central valued on R and there exist b,p,q∈U suchý ýthat H(x)=px for all x∈Rý, ýG(x)=bx for all x∈R with a(p−q)∈C;ý ýthere exist p,q∈U such that H(x)=px for all x∈Rý, ýG(x)=qx for all x∈R with ap=0;ý f(x1,…,xn)2 is central valued on R and there exist q∈Uý, ýλ∈C and an outer derivation g of Uý ýsuch that H(x)=xqλx−g(x) for all x∈Rý, ýG(x)=qx(x) for all x∈Rý, ýwith a∈C;ý
R satisfies s4 and there exist b,p∈U suchý ýthat H(x)=px for all x∈Rý, ýG(x)=bx for all x∈Rý.