Annihilating co-commutators with generalized skew derivations on multilinear polynomials

Let ℛ be a prime ring of characteristic different from 2, Q_r be its right Martindale quotient ring, Q be its two-sided Martindale quotient ring and C be its extended centroid. Suppose that ℱ, g are additive mappings from ℛ into itself and that f(x₁, …, x_n) is a non-central multilinear polynomial over C with n non-commuting variables. We prove the following results: (a) If ℱ and g are generalized derivations of ℛ such that (Formula presented.) for all (Formula presented.), then one of the following holds: (a) there exists q∈Q such that ℱ(x) = xq and g(x) = qx for all x∈ℛ. (b) there exist c,q∈Q such that ℱ(x) = qx+xc, g(x) = cx+xq for all x∈ℛ, and f(x₁, …, x_n)² is central-valued on ℛ. (b) If ℱ is a generalized skew derivation of ℛ such that (Formula presented.) for all (Formula presented.), then one of the following holds: (a) there exists λ∈C such that ℱ(x) = λx for all x∈ℛ; (b) there exist q∈Q_r and λ∈C such that ℱ(x) = (q+λ)x+xq for all x∈ℛ, and f(x₁, …, x_n)² is central-valued on ℛ.