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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 M be a prime Γ-ring, I a nonzero ideal, θ an automorphism and d a θ-derivation of M. In this article we have proved the following result: (1) If d([x, y]α) = ±([x, y]α) or d((x o y)α) = ±((x o y)α) for all x, y ε I;α ε Γ, then M is commutative. (2) Under the hypothesis dθ = θd and Char M, 2, if (d(x) o d(y))α = 0 or [d(x),d(y)]α = 0 for all x, y 2 I;α α Γ, then M is commutative. (3) If d acts as a homomorphism or an antihomomorphism on I, then d = 0 or M is commutative. Moreover, an example is given to demonstrate that the primeness imposed on the hypothesis of the various results is essential. [ABSTRACT FROM AUTHOR]

Let M be a prime Γ-ring, I a nonzero ideal, θ an automorphism and dd a θ-derivation of MM. In this article we have proved the following result: (1) If d([x,y]α)=±([x,y]α) or d((xoy)α)=±((xoy)α) for x,y∈I;α∈Γ, then M is commutative. (2) Under the hypothesis dθ=θd and CharM ≠ 2, if (d(x) o d(y))α=0 or [d(x),d(y)]α=0 for all x,y∈I;α∈Γ, then M is commutative. (3) If dd acts as a homomorphism or an anti-homomorphism on I, then d=0 or M is commutative. Moreover, an example is given to demonstrate that the primeness imposed on the hypothesis of the various results is essential. [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]