Annihilator Conditions with Generalized Derivations on Multi linear Polynomials

Let R be a prime ring of characteristic different from 2 with right Utumi quotient ring U and extended centroid C. Let g be a generalized derivation of R, f (x(1), ...,x) a multilinear polynomial over C, a is an element of R, and I a nonzero right ideal of R. Suppose that a[g(f(r(1), ...,r(n))), f(r(1), ...,r(n))] = 0 for all r(i) is an element of I and al not equal 0. Then either g(x) = a(1)x with (a(1) - gamma)I = 0 for some a(1) is an element of U and gamma is an element of C, or there exists an idempotent element e is an element of soc(RC) such that IC = eRC and one of the following holds: (i) f (x, ..., x) is central-valued in eRe; (ii) g(x) = bx + xc, where b,c is an element of U with (c - b - alpha)e = 0 for some alpha is an element of C and f (xi, ..., x(n))(2) is central-valued in eRe.