Pair of (generalized-)derivations on rings and banach algebras

Let n be a fixed positive integer, R be a 2n!-torsion free prime ring and μ, ν be a pair of generalized derivations on R. If (μ²(x)+ ν(x), xⁿ) = 0 for all x ∈ R, then μ and ν are either left multipliers or right multipliers. Let n be a fixed positive integer, R be a noncommutative 2n!-torsion free prime ring with the center C_R and d, g be a pair of derivations on R. If (d²(x) + g(x), xⁿ) ∈ C_R for all x ∈ R, then d = g = 0. Then we apply these purely algebraic techniques to obtain several range inclusion results of pair of (generalized-)derivations on a Banach algebra.