Annihilating co-commutators with generalized skew derivations on multilinear polynomials

Abstract

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 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 thatfor all , 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 is central-valued on ℛ.(b) If ℱ is a generalized skew derivation of ℛ such thatfor all , 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 is central-valued on ℛ.