Annihilating co-commutators with generalized skew derivations on multilinear polynomials

ABSTRACTLet ℛ be a prime ring of characteristic different from 2, 𝒬r 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,…,xn) 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 f(r¯)(ℱ(f(r¯))f(r¯)−f(r¯)𝒢(f(r¯)))=0 for all r¯∈ℛn, 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,…,xn)2 is central-valued on ℛ.(b) If ℱ is a generalized skew derivation of ℛ such that f(r¯)[ℱ(f(r¯)),f(r¯)]=0for all r¯∈ℛn, then one of the following holds:(a) there exists λ∈𝒞 such that ℱ(x) = λx for all x∈ℛ;(b) there exist q∈𝒬r and λ∈𝒞 such that ℱ(x) = (q+λ)x+xq for all x∈ℛ, and f(x1,…,xn)2 is central-valued on ℛ.