A NOTE ON CERTAIN CENTRAL DIFFERENTIAL IDENTITIES WITH GENERALIZED DERIVATIONS

Let R be a noncommutative prime ring of characteristic different from 2 with right Utumi quotient ring U and extended centroid C, I a nonzero right ideal of R. Let f(x1, . . . , xn) be a non-central multilinear polynomial over C, m ≥ 1 a fixed integer, a a fixed element of R, G a non-zero generalized derivation of R. If aG(f(r1, . . . , rn))ᵐ ∈ Z(R) for all r1, . . . , rn ∈ I, then one of the following holds:(1) aI = aG(I) = (0);(2) G(x) = qx, for some q ∈ U and aqI = 0;(3) [f(x1, . . . , xn), xn+1]xn+2 is an identity for I;(4) G(x) = cx + [q, x] for all x ∈ R, where c, q ∈ U such that cI = 0 and [q, I]I = 0;(5) dimC(RC) ≤ 4;(6) G(x) = αx, for some α ∈ C; moreover a ∈ C and f(x1, . . . , xn)ᵐ is central valued on R.