A note on generalized skew derivations on Lie ideals

Let \(\mathcal {R}\) be a prime ring, \(\mathcal {Z(R)}\) its center, \(\mathcal {C}\) its extended centroid, \(\mathcal {L}\) a Lie ideal of \(\mathcal {R}, \mathcal {F}\) a generalized skew derivation associated with a skew derivation d and automorphism \(\alpha \). Assume that there exist \(t\ge 1\) and \(m,n\ge 0\) fixed integers such that \( vu = u^m\mathcal {F}(uv)^tu^n\) for all \(u,v \in \mathcal {L}\). Then it is shown that either \(\mathcal {L}\) is central or \(\mathrm{char}(\mathcal {R})=2, \mathcal {R}\subseteq \mathcal {M}_2(\mathcal {C})\), the ring of \(2\times 2\) matrices over \(\mathcal {C}, \mathcal {L}\) is commutative and \(u^2\in \mathcal {Z(R)}\), for all \(u\in \mathcal {L}\). In particular, if \(\mathcal {L}=[\mathcal {R,R}]\), then \(\mathcal {R}\) is commutative.     

Generalized skew derivations with nilpotent values on Lie ideals

Let R be a prime ring and L a noncommutative Lie ideal of R. Suppose that f is a right generalized β-derivation of R associated with a β-derivation δ such that f(x) ⁿ  = 0 for all ${x\in L}$ , where n is a fixed positive integer. Then f = 0.     

Generalized skew derivations with nilpotent values on Lie ideals

Let R be a prime ring and L a noncommutative Lie ideal of R. Suppose that f is a right generalized β-derivation of R associated with a β-derivation δ such that f(x) ⁿ  = 0 for all x ? L{xin L}, where n is a fixed positive integer. Then f = 0.     

A result concerning nilpotent values with generalized skew derivations on Lie ideals

ABSTRACT

Let n≥1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that (F(x)F(y)?yx)ⁿ = 0 for all x,y∈L, then char(R) = 2 and R?M₂(C), the ring of 2×2 matrices over C.     

On Lie ideals with generalized derivations

Let R be a prime ring with characteristic different from 2, let U be a nonzero Lie ideal of R, and let f be a generalized derivation associated with d. We prove the following results: (i) If a ∈ R and [a, f(U)] = 0 then a ∈ Z or d(a) = 0 or U ? Z; (ii) If f ²(U) = 0 then U ? Z; (iii) If u ² ∈ U for all u ∈ U and f acts as a homomorphism or antihomomorphism on U then either d = 0 or U ? Z.     

Generalized derivations and left multipliers on Lie ideals

Let m, n be two fixed positive integers and let R be a 2-torsion free prime ring, with Utumi quotient ring U and extended centroid C. We study the identity F(x ^m+n+1) = F(x)x ^m+n  + x ^m D(x)x ⁿ for x in a non-central Lie ideal of R, where both F and D are generalized derivations of R and then determine the relationship between the form of F and that of D. In particular the conclusions of the main theorem say that if D is the non-zero map in R, then R satisfies the standard identity s ₄(x ₁, . . . , x ₄) and D is a usual derivation of R.     

Cocentralizing derivations and nilpotent values on Lie ideals

Let R be a prime ring with char R ≠ 2, L a non-central Lie ideal of R, d, g non-zero derivations of R, n ≥ 1 a fixed integer. We prove that if (d(x)x − xg(x)) ⁿ = 0 for all x ∈ L, then either d = g = 0 or R satisfies the standard identity s ₄ and d, g are inner derivations, induced respectively by the elements a and b such that a + b ∈ Z(R).     

Strong commutativity preserving generalized derivations on Lie ideals

We apply elementary matrix computations and the theory of differential identities to prove the following: let R be a prime ring with extended centroid C and L a noncommutative Lie ideal of R. Suppose that f?:?L?→?R is a map and g is a generalized derivation of R such that [f(x),?g(y)]?=?[x,?y] for all x,?y?∈?L. Then there exist a nonzero α?∈?C and a map μ?:?L?→?C such that g(x)?=?αx for all x?∈?R and f(x)?=?α^?1 x?+?μ(x) for all x?∈?L, except when R???M ₂(F), the 2?×?2 matrix ring over a field F.