Generalized Skew Derivations on Lie Ideals

In [17] Lee and Shiue showed that if R is a non-commutative prime ring, I a nonzero left ideal of R and d is a derivation of R such that [d(x)x, x]k = 0 for all x ∈ I, where k,m, n, r are fixed positive integers, then d = 0 unless R ∼= M2(GF (2)). Later in [1] Argaç and Demir proved the following result: Let R be a non-commutative prime ring, I a nonzero left ideal of R and k,m, n, r fixed positive integers. If there exists a generalized derivation g of R such that [g(x)x, x]k = 0 for all x ∈ I, then there exists a ∈ U , the left Utumi quotient ring of R, such that g(x) = xa for all x ∈ R, except when R ∼= M2(GF (2)) and I[I, I] = 0.