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Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring, L a non-central Lie ideal of R (i.e., L , R ≠ 0 ), n ≥ 1 a fixed integer, F and G two generalized skew derivations of R, associated with the same automorphism α of R. If (F (x y) − G (x) y) n = 0 , for any x , y ∈ L , then there exists a ∈ Q r such that , for any x ∈ R , unless when R satisfies the standard polynomial identity s 4 (x 1 , ... , x 4) . [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2, Z(R) its center, L a Lie ideal of R, and m, n, s, t ≥ 1 fixed integers with t ≤ m + n + s. Suppose that α is a non-trivial automorphism of R and let Φ(x, y) = [x, y]t – [x, y]m [α([x, y]),[x, y]]n [x, y]s. Thus, (a) if Φ(u, v) = 0 for any u, v ∈ L, then L ⊆ Z(R); (b) if Φ(u, v) ∈ Z(R) for any u, v ∈ L, then either L ⊆ Z(R) or R satisfies s4, the standard identity of degree 4. We also extend the results to semiprime rings. [ABSTRACT FROM AUTHOR]

Let R be a prime ring, I a nonzero ideal of R, and a ε R. Suppose that σ is a nontrivial automorphism of R such that af{(σ(x 0 y))n - (x 0 y)m} = 0 or a{(σ([x; y]))n - ([x; y])m} = 0 for all x,y ε I, where n and m are fixed positive integers. We prove that if char(R) > n + 1 or char(R) = 0, then either a = 0 or R is commutative. [ABSTRACT FROM AUTHOR]