Derivations In Prime Rings Research Articles

An Engel condition with two generalized derivations in prime rings

Let R be a noncommutative prime ring with extended centroid C and maximal right ring of quotients Q. Let I be a nonzero ideal of R, and let g, h be two generalized derivations of R. Suppose that for all where are fixed positive integers. Then, one of the following conditions holds: there exist such that and for all m = s, n = t and there exist and such that g(x) = xa and for all C is a finite field, the matrix ring over C for some integer and there exist such that g(x) = xa and h(x) = ux for all

Annihilators of power values of b-generalized derivations in prime rings

Let $R$ be a prime ring with extended centroid $C$ and maximal left ring of quotients $Q_{ml}(R)$. For a nonzero element $b\in R$ let $F:R\rightarrow R$ be a right generalized $b$-derivation associated with the map $d$ of $R$. Suppose that $s\left(F(x)\right)^n=0$ for all $x\in R$ where $s$ is a nonzero element in $R$ and $n\geq 1$ is a fixed positive integer. Then there exist some $c\in Q{ml}(R)$ and $\beta \in C$ such that $d(x)=ad_c(x)$, $F(x)=(b+\beta)xb$ for all $x\in R$ and either $s(c+\beta)=0$ or $b(c+\beta)=0$.

Multiplicative (generalized)-derivations of prime rings that act as $n$-(anti)homomorphisms

Let R be a prime ring. In this note, we describe the possible forms of multiplicative (generalized)-derivations of R that act as n-homomorphism or n-antihomomorphism on nonzero ideals of R. Consequently, from the given results one can easily deduce the results of Gusić ([7]).

A Characterization of Derivations in Prime Rings with Involution

The purpose of this paper is to investigate $*$-differential identities satisfied by pair of derivations on prime rings with involution. In particular, we prove that if a 2-torsion free noncommutative ring $R$ admit nonzero derivations $d_1, d_2$ such that $[d_1(x), d_2(x^*)]=0$ for all $x\in R$, then $d_1=\lambda d_2$, where $\lambda\in C$. Finally, we provide an example to show that the condition imposed in the hypothesis of our results are necessary.

A Note on Multiplicative (Generalized) (α, β)-Derivations in Prime Rings

Abstract Let R be a prime ring with center Z(R). A map G : R →R is called a multiplicative (generalized) (α, β)-derivation if G(xy)= G(x)α(y)+β(x)g(y) is fulfilled for all x; y ∈ R, where g : R → R is any map (not necessarily derivation) and α; β : R → R are automorphisms. Suppose that G and H are two multiplicative (generalized) (α, β)-derivations associated with the mappings g and h, respectively, on R and α, β are automorphisms of R. The main objective of the present paper is to investigate the following algebraic identities: (i) G(xy) + α(xy) = 0, (ii) G(xy) + α(yx) = 0, (iii) G(xy) + G(x)G(y) = 0, (iv) G(xy) = α(y) ○ H(x) and (v) G(xy) = [α(y), H(x)] for all x, y in an appropriate subset of R.

Weak Jordan derivations of prime rings

Let R be a prime ring with extended centroid C and with maximal left ring of quotients . An additive map is called a weak Jordan derivation if for all . Applying the theory of functional identities and dealing with the low dimensional cases, we give a complete characterization of weak Jordan derivations of prime rings. Moreover, we generalize Brešar's theorem concerning additive maps satisfying for all .

On multiplicative generalized (a, b)-derivations in prime rings

Let R be an associative ring with center Z(R). A mapping F:R → R is called a multiplicative generalized derivation on R if there exists a derivation d on R such that F(xy)=F(x)y+xd(y) for all x,y ∈ R. Let α, β be endomorphisms of R. A mapping F:R → R (not necessarily additive) is said to be a multiplicative generalized (α, β)-derivation if there exists an (α, β)-derivation d on R such that F(xy)=F(x)α(y)+β(x)d(y) holds for all x,y ∈ R. In this paper we investigate some identities involving multiplicative generalized (α, β)-derivation in a prime ring R and obtain commutativity of R.

ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS

Let $R$ be a ring, a mapping $F:R\rightarrow R$ together with a mapping $d:R\rightarrow R$ is called a multiplicative (generalized)-reverse derivation if $F(xy)=F(y)x+yd(x)$ for all $x,y\in R$. The aim of this note is to investigate the commutativity of prime rings admitting multiplicative (generalized)-reverse derivations. Precisely, it is proved that for some nonzero element $a$ in $R$ the conditions: $a(F(xy)\pm xy)=0$, $a(F(x)F(y)\pm xy)=0$, $a(F(xy)\pm F(y)F(x))=0$, $a(F(x)F(y)\pm yx)=0$, $a(F(xy)\pm yx)=0$ are sufficient for the commutativity of $R$. Moreover, we describe the possible forms of generalized reverse derivations of prime rings.

Engel type identities with generalized derivations in prime rings

Let [Formula: see text] be a noncommutative prime ring with its Utumi ring of quotients [Formula: see text], [Formula: see text] the extended centroid of [Formula: see text], [Formula: see text] a generalized derivation of [Formula: see text] and [Formula: see text] a nonzero ideal of [Formula: see text]. If [Formula: see text] satisfies any one of the following conditions: (i) [Formula: see text], [Formula: see text], [Formula: see text], (ii) [Formula: see text], where [Formula: see text] is a fixed integer, then one of the following holds: (1) there exists [Formula: see text] such that [Formula: see text] for all [Formula: see text]; (2) [Formula: see text] satisfies [Formula: see text] and there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text]; (3) char [Formula: see text], [Formula: see text] satisfies [Formula: see text] and there exist [Formula: see text] and an outer derivation [Formula: see text] of [Formula: see text] such that [Formula: see text] for all [Formula: see text].

On a Functional Equation Related to Jordan Triple Derivations in Prime Rings

A classical result of Herstein asserts that any Jordan derivation on a prime ring with char(R) ≠ 2 is a derivation. It is our aim in this paper to prove the following result, which is in the spirit of Herstein’s theorem. Let R be a prime ring with char(R) = 0 or char(R) > 4, and let D: R → R be an additive mapping satisfying the relation D(x4) = D(x)x3 + xD(x2)x + x3D(x) for all x ∈ R. In this case, D is a derivation.

On Lie ideals with generalized $$(\alpha , \alpha )$$-derivations in prime rings

Let R be a prime ring and $$\alpha $$ an automorphism on R. An additive mapping F on R is said to be a generalized $$(\alpha , \alpha )$$ -derivation on R if there exists an $$(\alpha , \alpha )$$ -derivation d on R such that $$F(xy)=F(x)\alpha (y)+\alpha (x)d(y)$$ holds for all $$x, y\in R$$ . In this paper our main objective is to study the following identities: (i) $$G(xy) \pm F(x)F(y)\in Z(R);$$ (ii) $$G(xy)\pm F(x)F(y)\pm \alpha (yx) =0;$$ (iii) $$G(xy) \pm F(x)F(y)\pm \alpha (xy)\in Z(R);$$ (iv) $$G(xy)\pm F(x)F(y)\pm \alpha ([x, y]) =0;$$ (v) $$G(xy)\pm F(x)F(y)\pm \alpha (x\circ y)=0;$$ for all x, y in some suitable subset of R, where G and F are two generalized $$(\alpha ,\alpha )$$ -derivations on R

An identity with derivations in prime rings

An identity with derivations in prime rings

On some functional equation arising from $(m,n)$-Jordan derivations of prime rings

On some functional equation arising from $(m,n)$-Jordan derivations of prime rings

Annihilator of Power Values of Generalized Derivations in Prime Rings

Let R be a prime ring of characteristic different from 2, U its Utumi quotient ring, C the center of U, F a non-zero generalized derivation of R and L a non-commutative Lie ideal of R. Suppose that there exists \(0\ne a\in R\) such that \(a(u^s[F(u),u]u^t)^n=0\) for all \(u \in L\), where \(s\ge 0, t\ge 0, n\ge 1\) are fixed integers. Then either \(F(x)=\alpha x\) for all \(x\in R\) with \(\alpha \in C\) or R satisfies \(s_4(x_1,\ldots ,x_4)\), the standard identity in four variables, and \(F(x)=bx+xb+\alpha x\) for all \(x\in R\), for some \(b\in U\) and \(\alpha \in C\).

English

English

Jordan [! \large !]$\sigma $-derivations of prime rings

Let $R$ be a noncommutative prime ring with extended centroid~$C$ and with $Q_{mr}(R)$ its maximal right ring of quotients. From the viewpoint of functional identities, we give a complete characterization of Jordan $\sigma $-derivations of $R$ with $\sigma $ an epimorphism. Precisely, given such a Jordan $\sigma $-derivation $\de \colon R\to Q_{mr}(R)$, it is proved that either $\delta $ is a $\sigma $-derivation or a derivation $d\colon R\to Q_{mr}(R)$ and a unit $u\in Q_{mr}(R)$ exist such that $\delta (x)=ud(x)+\mu (x)u$ for all $x\in R$, where $\mu \colon R\to C$ is an additive map satisfying $\mu (x^2)=0$ for all $x\in R$. In addition, if $\sigma $ is an X-outer automorphism, then $\delta $ is always a $\sigma $-derivation.

A note on generalized Lie derivations of prime rings

Let R be a prime ring of characteristic not 2, A be an additive subgroup of R, and F, T, D, K: A-R be additive maps such that F([x, y]) = F(x)y-yK(x)-T(y)x + xD(y) for all x, yEA. Our aim is to deal with this functional identity when A is R itself or a noncentral Lie ideal of R. Eventually, we are able to describe the forms of the mappings F, T, D, and K in case A = R with deg(R) > 3 and also in the case A is a noncentral Lie ideal and deg(R) > 9. These enable us in return to characterize the forms of both generalized Lie derivations, D-Lie derivations and Lie centralizers of R under some mild assumptions. Finally, we give a generalization of Lie homomorphisms on Lie ideals.

On $m$-commuting mappings with skew derivations in prime rings

On $m$-commuting mappings with skew derivations in prime rings

On Functional Equation Related to a Class of Generalized Inner Derivations in Prime Rings

The main purpose of this paper is to prove the following result. Let R be a prime ring of \(\mathrm{char}(R)=0\) or \(\mathrm{char}(R)>4,\) and let \(T{\text {:}}\,R\rightarrow R\) be an additive mapping satisfying the relation \(T(x^{4})=T(x)x^{3}-xT(x^{2})x+x^{3}T(x)\) for all \(x\in R.\) In this case, T is of the form \(4T(x)=qx+xq\) for all \(x\in R,\) where q is some fixed element from the symmetric Martindale ring of quotients of R. Some functional equations related to bicircular projections are also investigated.

Engel-Type Conditions Involving Two Generalized Skew Derivations in Prime Rings

Let ℛ be a prime ring of characteristic different from 2, 𝒬r the right Martindale quotient ring of ℛ, 𝒞 the extended centroid of ℛ, F, G two generalized skew derivations of ℛ, and k ≥ 1 be a fixed integer. If [F(r), r]kr − r[G(r), r]k = 0 for all r ∈ ℛ, then there exist a ∈ 𝒬r and λ ∈ 𝒞 such that F(x) = xa and G(x) = (a + λ)x, for all x ∈ ℛ.