Derivations vanishing on commutator identity involving generalized derivation on multilinear polynomials in prime rings

AbstractLet R be a prime ring of characteristic different from 2 with its Utumi quotient ring U and extended centroid C, f(x1,…,xn) be multilinear polynomial over C, which is not central valued on R. If d is a non-zero derivation of R and F is a non-zero generalized derivation of R such that d([F(f(r1,…,rn))f(r1,…,rn),F(f(s1,…,sn))f(s1,…,sn)])=0for all r1,…,rn,s1,…,sn∈R, then there exists c∈U such that F(x) = cx, for all x∈R and one of the following holds: (1) f(x1,…,xn)2 is central valued on R;(2) R satisfies s4, the standard identity of degree 4.