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

Let R be a prime ring of characteristic different from 2 with its Utumi quotient ring U and extended centroid C, (Formula presented.) 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 (Formula presented.) for all (Formula presented.), then there exists (Formula presented.) such that F(x) = cx, for all (Formula presented.) and one of the following holds: (1) (Formula presented.) is central valued on R; (2) R satisfies s 4 , the standard identity of degree 4. © 2018, © 2018 Taylor & Francis Group, LLC.