Let A be an algebra over the real or complex field F. An additive mapping d : A → A is said to be a left derivation resp., derivation if the functional equation d xy xd y yd x resp., d xy xd y d x y holds for all x, y ∈ A. Furthermore, if the functional equation d λx λd x is valid for all λ ∈ F and all x ∈ A, then d is a linear left derivation resp., linear derivation . An additive mapping G : ...

Let n ∈ N − {1}, and let A be a Banach algebra. An additive map D : A → A is called n-Jordan derivation if D(a) = D(a)a + aD(a)a + ...+ aD(a)a+ aD(a), for all a ∈ A. Using fixed point methods, we investigate the stability of n–Jordan derivations (n–Jordan ∗−derivations) on Banach algebras (C∗−algebras). Also we show that to each approximate ∗−Jordan derivation f in a C∗− algebra there correspon...

Let A be an algebra, and let θ, φ be ring automorphisms of A. An additive mapping H : A → A is called a θ, φ -derivation if H xy H x θ y φ x H y for all x, y ∈ A. Moreover, an additive mapping F : A → A is said to be a generalized θ, φ -derivation if there exists a θ, φ derivation H : A → A such that F xy F x θ y φ x H y for all x, y ∈ A. In this paper, we investigate the superstability of gene...

In this paper, we introduced the concept of f -derivation which is a generalization of derivation in subtraction algebra, and some related properties are investigated.

abstract. let r be a 2-torsion free ring with identity. in this paper, first we prove that any jordan left derivation (hence, any left derivation) on the full matrix ringmn(r) (n  2) is identically zero, and any generalized left derivation on this ring is a right centralizer. next, we show that if r is also a prime ring and n  1, then any jordan left derivation on the ring tn(r) of all n×n up...

LetN be a zero-symmetric left near-ring, not necessarily with amultiplicative identity element; and letZ be its multiplicative center. DefineN to be 3-prime if for all a, b ∈ N\{0}, aNb / {0}; and callN 2-torsion-free if N, has no elements of order 2. A derivation onN is an additive endomorphism D of N such that D xy xD y D x y for all x, y ∈ N. A generalized derivation f with associated deriva...

let a be a unital r-algebra and m be a unital a-bimodule. it is shown that every jordan derivation of the trivial extension of a by m, under some conditions, is the sum of a derivation and an antiderivation.

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...

Let $f(X_1,\dots, X_n)$ be a nonzero multilinear noncommutative polynomial. If $A$ is unital algebra with surjective inner derivation, then every element in can written as $f(a_1,\dots,a_n)$ for some $a_i\in A$.