Generalized derivations as homomorphisms or anti-homomorphisms on Lie ideals

Generalized Skew Derivations on Lie Ideals

In [17] Lee and Shiue showed that if R is a non-commutative prime ring, I a nonzero left ideal of R and d is a derivation of R such that [d(x)x, x]k = 0 for all x ∈ I, where k,m, n, r are fixed positive integers, then d = 0 unless R ∼= M2(GF (2)). Later in [1] Argaç and Demir proved the following result: Let R be a non-commutative prime ring, I a nonzero left ideal of R and k,m, n, r fixed posi...

Stability of Homomorphisms and Generalized Derivations on Banach Algebras

Approximate n-Lie Homomorphisms and Jordan n-Lie Homomorphisms on n-Lie Algebras

and Applied Analysis 3 Park and Rassias 59 proved the stability of homomorphisms in C∗-algebras and Lie C∗-algebras and also of derivations on C∗-algebras and Lie C∗-algebras for the Jensen-type functional equation μf ( x y 2 ) μf ( x − y 2 ) − fμx 0 1.6 for all μ ∈ T1 : {λ ∈ C; |λ| 1}. In this paper, by using the fixed-point methods, we establish the stability of n-Lie homomorphisms and Jordan...

Notes on Generalized Derivations on Lie Ideals in Prime Rings

Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that usH(u)ut = 0 for all u ∈ L, where s ≥ 0, t ≥ 0 are fixed integers. Then H(x) = 0 for all x ∈ R unless char R = 2 and R satisfies S4, the standard identity in four variables. Let R be an associative ring with center Z(R). For x, y ∈ R, the commutator xy− yx will be denoted by [x, y]. An add...

Lie Ideals and Generalized Derivations in Semiprime Rings

Let R be a 2-torsion free ring and L a Lie ideal of R. An additive mapping F : R ! R is called a generalized derivation on R if there exists a derivation d : R to R such that F(xy) = F(x)y + xd(y) holds for all x y in R. In the present paper we describe the action of generalized derivations satisfying several conditions on Lie ideals of semiprime rings.