Annihilating co-commutators with generalized skew derivations on multilinear polynomials

Let 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 Q be its extended centroid. Suppose that F, Q are additive mappings from R into itself and that f(x(1), ...,x(n))f(x(1), ...,x(n)) is a non-central multilinear polynomial over Q with n non-commuting variables. We prove the following results: (a) If F and Q are generalized derivations of R such that f(r)(F(f(r))f(r)-f(r)G(f(r)))=0f(r)(F(f(r))f(r)-f(r)Q(f(r)))=0 for all r Rnr Rn, then one of the following holds: (a) there exists qQ such that F(x) = xq and Q(x) = qx for all x R. (b) there exist c,qQ such that F(x) = qx+xc, Q(x) = cx+xq for all x R, and f(x(1), ...,x(n))2f(x(1), ...,x(n))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)]=0f(r)[F(f(r)),f(r)]=0 for all r Rnr Rn, then one of the following holds: (a) there exists QQ such that F(x) = x for all x R; (b) there exist qQ r and QQ such that F(x) = (q+ )x+xq for all x R, and f(x(1), ...,x(n))2f(x(1), ...,x(n))2 is central-valued on R.