Let R be a prime ring of characteristic different from 2, Q_r its right Martindale quotient ring, and C its extended centroid. Suppose that F, G are generalized skew derivations of R, with the same associated automorphism, and f(x₁,…, x_n) a noncentral multilinear polynomial over C with n noncommuting variables, such that for all r₁,…, r_n ∈ R. Then we describe all possible forms of F and G.

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, F a non-zero generalized derivation of R. Suppose that [F(u), u]F(u) = 0 for all u ε L, then one of the following holds: there exists α ε C such that F(x) = α x for all x ε R R satisfies the standard identity s ₄ and there exist a ε U and α ε C such that F(x) = ax + xa + αx for all x ε R. We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.

Let R be a non-commutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, L a non-central Lie ideal of R, G a non-zero generalized derivation of R. If [G(u), u]ⁿ = [G(u), u], for all u ∈ L, with n > 1, then one of the following holds: (1) R satisfies the standard identity s₄(x₁, . . . , x₄) and there exist a ∈ U and α ∈ C such that G(x) = ax + xa + αx for all x ∈ R; (2) there exists γ ∈ C such that G(x) = γx for all x ∈ R.

Let R be a non-commutative prime ring of characteristic different from 2, Q_r its right Martindale quotient ring and C its extended centroid. Suppose that L is a non-central Lie ideal of R, F a non-zero generalized skew derivation of R and $n\ge 1$ a fixed integer. If $${(F(x)°F(y))}^n={(x°y)}^n$$ for all $x,y\in L,$ then one of the following holds: