Generalized Jordan derivations on prime rings and standard operator algebras

In this paper we initiate the study of generalized Jordan derivations and generalized Jordan triple derivations on prime rings and standard operator algebras

Generalized derivations on ideals of prime rings

WOS: 000328081500001Let R be a prime ring. By a generalized derivation we mean an additive mapping g : R -> R such that g(xy) = g(x)y + xd(y) for all x, y is an element of R where d is a derivation of R. In the present paper our main goal is to generalize some results concerning derivations of prime rings to generalized derivations of prime rings

A note on generalized derivations of prime rings

We show that a generalized derivation on a prime ring, that acts as a homomorphism or an anti-homomorphism on a non-zero ideal in the ring, is the zero map or the identity map

On Maps of Period 2 on Prime and Semiprime Rings

A map f of the ring R into itself is of period 2 if f2x=x for all x∈R; involutions are much studied examples. We present some commutativity results for semiprime and prime rings with involution, and we study the existence of derivations and generalized derivations of period 2 on prime and semiprime rings

On Commutativity of Rings with Generalized Derivations

The concept of derivations as well as of generalized inner derivations have been generalized as an additive function F : R &#8594; R satisfying F(xy) = F(x)y + xd(y) for all x, y &#8712; R, where d is a derivation on R, such a function F is said to be a generalized derivation. In the present paper we have discussed the commutativity of prime rings admitting a generalized derivation F satisfying (i) [F(x), x] = 0, (ii) F([x, y]) = [x, y], and (iii) F(x &#9702; y) = x &#9702; y for all x, y in some appropriate subset of R.</p

On commutativity of prime and semiprime rings with generalized derivations

Let $R$ be a prime ring, extended centroid $C$ and $m, n, k \geq1$ are fixed integers. If $R$ admits a generalized derivation $F$ associated with a derivation $d$ such that $(F(x)\circ y)^{m}+(x\circ d(y))^{n}=0$ or $(F(x)\circ_{m} y)^{k} + x\circ_{n} d(y)$=0 for all $x, y \in I$, where $I$ is a nonzero ideal of $R$, then either $R$ is commutative or there exist $b\in U$, Utumi ring of quotient of $R$ such that $F(x)=bx$ for all $x \in R$. Moreover, we also examine the case $R$ is a semiprime ring

On Centralizing and Generalized Derivations Of prime Rings with involution

&nbsp;Let (R,∗) be a 2-torsion free ∗-prime ring with involution ∗, L= 0 be a nonzero square closed ∗-Lie ideal of R and Z the center of R. An additive mapping F: R −→ R is called a generalized derivation on R if there exists a derivation d: R−→Rcommutes with ∗ such that F(xy) = F(x)y +xd(y) holds for all x,y ∈ R. In the present paper, we shall show that L is contained in the center of R such that R admits a generalized derivations F and G with associated derivations d and g commute with ∗ satisfying several conditions

A result on generalized derivations on right ideals of prime rings

Let R be a prime ring of characteristic other than 2 and let I be a nonzero right ideal of R. Also let U be the right Utumi quotient ring of R and let C be the center of U. If G is a generalized derivation of R such that [[G(x), x], G(x)] = 0 for all x ∈ I, then R is commutative or there exist a, b ∈ U such that G(x) = ax + xb for all x ∈ R and one of the following assertions is true: (1) (a - λ)I = (0) = (b + λ)I for some λ ∈ C, (2) (a - λ)I = (0) for some λ ∈ C and b ∈ C.Нехай R — просте кiльце, характеристика якого не дорiвнює 2, а I — ненульовий правий iдеал R. Нехай U — праве фактор-кiльце Утумi кiльця R, а C — центр U. Якщо G є узагальненим диференцiюванням R таким, що [[G(x),x],G(x)]=0 для всiх x∈I, то R є комутативним або iснують a,b∈U такi, що G(x)=ax+xb для всiх x∈R i виконується одне з наступних тверджень: (1)(a−λ)I=(0)=(b+A)Iдля деякогоλ∈C, (2)(a−λ)I=(0)для деякогоλ∈Cіb∈C