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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 (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]