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Let R be a prime ring with char(R)≠2. A biadditive symmetric map Δ: R×R → R is called symmetric (α, β)-biderivation if, for any fixed y ∈ R, the map x ↦ Δ(x,y) is a (α, β)-derivation. A symmetric biadditive map Γ : R×R → R is a symmetric generalized (α, β)-biderivation if for any fixed y ∈ R, the map x ↦ Γ(x,y) is a generalized (α, β)-derivation of R associated with the (α, β)-derivation Δ(.,y). In the present paper, we investigate the commutativity of a ring having a generalized (α, β)-biderivation satisfying certain algebraic conditions. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with center Z(R), and d a derivation of R. Suppose that (d[x, y]k)n - m[x, y]k ∈ Z(R) for all x, y ∈ R, where m≠ n, k ≥ 1 are fixed integers. Then d = 0 or R satisfies s4, the standard identity in four variables. In the case (d[x, y]kn - m[x, y]k = 0 for all x, y ∈ R, then d = 0 or R is commutative. [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]

Let (R, *) be a 2-torsion free *-prime ring with involution * and center Z(R), U a nonzero square closed *-Lie ideal of R. An additive mapping F : R→R is called a generalized derivation if there exits a derivation d : R→R such that F(xy)=F(x)y+xd(y). In the present paper, we prove that U ⊆ Z(R) if any one of following conditions holds: 1) [F (u), u] = 0, 2) [d(u), F(v) = 0, 3) d(u)oF(v) = 0; 4) [d(u), F(v)] = ±[u,v], 5) d(u)oF(v) = ±uov, 6) d(u)F(v)±uv ϵ Z(R), for all u, v ϵ U. Furthermore, an example is given to demonstrate that the *-primeness hypothesis is not superfluous. [ABSTRACT FROM AUTHOR]

Let N be a prime near-ring and σ, τ be automorphisms of N. Suppose that F is a nonzero left generalized (σ, σ) -derivation of N associated with a nonzero (σ, σ)-derivation d. In the present paper, our objective is to prove that N is a commutative ring if one of the following conditions holds: (i) F([x, y]) = 0, (ii) F(x o y) = 0, (iii) F([x,y]) = σ([x, y]), (iv) F([x,y]) = σ(-xy + yx) for all x,y ∈ N. Further, suppose that F is a left generalized (σ, τ)-derivation of N associated with a nonzero (σ, τ)-derivation d. If F acts as a homomorphism or an anti-homomorphism on N, then F = τ. Finally, an example is given to demonstrate that the hypothesis of primeness in the above results is essential. [ABSTRACT FROM AUTHOR]