Derivations In Semiprime Rings Research Articles

On the Multiplicative (Generalised) (α, α)−Derivations of Semiprime Rings

The algebraic properties and identities of a semiprime ring are investigated with the help of the multiplicative (generalised)-(α, α)-reverse derivation on the non-empty ideal of the semiprime ring.

Some Identities Involving Multiplicative (Generalized)(α; 1)-Derivations in Semiprime Rings

Let R be a semiprime ring, I a non-zero ideal of R and α be an automorphism of R. A map F : R to R is said to be a multiplicative (generalized)(α, 1)-derivation associated with a map d : R to R such that F (xy) = F (x) α (y) + xd (y), for all x, y in R. In the present paper, we shall prove that R contains a non-zero central ideal if any one of the following holds: (i) F [x, y] ± [x,y] =0; (ii) F (xoy)±α(xoy) = 0; (iii) F [x, y] = [F (x) , y]α;1 ; (iv) F [x, y] =(F (x) oy)α;1 ; (v) F (xoy) = [F(x) , y]α;1 and (vi)F (xoy) = (F (x) oy)α;1, for all x, y in I.

Functional equations related to higher derivations in semiprime rings

Abstract We investigate the additivity and multiplicativity of centrally extended higher derivations and show that every centrally extended higher derivation of a semiprime ring with no nonzero central ideals is a higher derivation. Moreover, we study preservation of the center of the ring by a centrally extended higher derivation.

Generalised Inner Derivations in Semi Prime Rings

Let A be any ring and f(xy) = f(x)y+xha(y), where f be any generalised inner derivation(G.I.D ) a be the fixed element of A. In this paper, it is shown that (i) ha must necessarily be a derivation for semi prime ring A. (ii) ∃ no generalized inner derivations f : A → A such that f(x ◦ y) = x ◦ y or f(x ◦ y) + x ◦ y = 0 ∀ x,y ∈ A, We have proved Havala [2] def. p.1147, Herstein [3] Lemma 3.1 p. 1106 as corollaries, along with other results.

Comparison of Jordan (sigma,tau)- Derivations and Jordan Triple (sigma,tau)- Derivations in Semiprime Rings

Let R be a 3!-torsion free semiprime ring, τ, σ two endomorphisms of R, d:R→R an additive mapping and L be a noncentral square-closed Lie ideal of R . An additive mapping d:R→R is said to be a Jordan (σ,τ)-derivation if d(x²)=d(x)σ(x)+τ(x)d(x) holds for all x,y∈R. Also, d is called a Jordan triple (σ,τ)-derivation if d(xyx)=d(x)σ(yx)+τ(x)d(y)σ(x)+τ(xy)d(x), for all x,y∈R. In this paper, we proved the following result: d is a Jordan (σ,τ)-derivation if and only if d is a Jordan triple (σ,τ)-derivation.

On centrally-extended multiplicative (generalized)-$(\alpha,\beta)$-derivations in semiprime rings

Let $R$ be a ring with center $Z$ and $\alpha$, $\beta$ and $d$ mappings of $R$. A mapping $F$ of $R$ is called a centrally-extended multiplicative (generalized)-$(\alpha,\beta)$-derivation associated with $d$ if $F(xy)-F(x)\alpha(y)-\beta(x)d(y)\in Z$ for all $x, y \in R$. The objective of the present paper is to study the following conditions: (i) $F(xy)\pm \beta(x)G(y)\in Z$, (ii) $F(xy)\pm g(x)\alpha(y)\in Z$ and (iii) $F(xy)\pm g(y)\alpha(x)\in Z$ for all $x,y$ in some appropriate subsets of $R$, where $G$ is a multiplicative $($generalized$)$-$(\alpha,\beta)$-derivation of $R$ associated with the map $g$ on $R$.

Commutativity with derivations of semiprime rings

Commutativity with derivations of semiprime rings

On generalized derivations in semiprime rings involving anticommutator

Let $$\mathfrak {R}$$ be a prime ring with characteristic different from 2 and m, n, k be fixed positive integers. In this paper we study the case when $$\mathfrak {R}$$ admits a generalized derivation $$\mathscr {F}$$ with associated derivation $$\mathfrak {D}$$ such that $$(i)~(\mathscr {F}(x)\circ \mathscr {F}(y))^k=\mathscr {F}(x\circ _ky)~(ii)~ \mathscr {F}(x)\circ _m \mathscr {F}(y) =(\mathscr {F}(x\circ y))^n~(iii)~(\mathscr {F}\left( x\right) \circ \mathscr {F}\left( y\right) )^m =(\mathscr {F}(x\circ y))^n,$$ for all $$x,y \in \mathscr {I}$$ , where $$ \mathscr {I}$$ is a non-zero ideal of $$\mathfrak {R}$$ . Moreover, we also examine the case when $$\mathfrak {R}$$ is a semiprime ring.

Centralizers and Jordan triple derivations of semiprime rings

Let R be a semiprime ring with extended centroid C and with maximal left ring of quotients . An additive map is called a Jordan triple derivation if for all . In 1957, Herstein proved that a Jordan triple derivation, which is also a Jordan derivation, of a noncommutative prime ring of characteristic 2, must be a derivation. In 1989, Brešar proved that any Jordan triple derivation of a 2-torsion free semiprime ring is a derivation. In the article, we give a complete characterization of Jordan triple derivations of arbitrary semiprime rings. To get such a characterization we first show that, in some sense, an additive map satisfying for all can be realized as a centralizer with only an exceptional case that and R is commutative.

Left Multiplicative Generalized Jordan Derivations of Semiprime Rings

In this paper we prove that left multiplicative generalized Jordan derivation and left multiplicative generalized Jordan triple derivation of 2-torsion free semiprime rings are left multiplicative generalized derivation.

On multiplicative (generalized)-derivations in semiprime rings

On multiplicative (generalized)-derivations in semiprime rings

Strong Commutativity Preserving Skew Derivations in Semiprime Rings

Let R be a semiprime ring of characteristic different from 2, C its extended centroid, Z(R) its center, F and G non-zero skew derivations of R with associated automorphism \(\alpha \) and m, n positive integers such that $$\begin{aligned}{}[F(x),G(y)]_m=[x,y]^n ~\mathrm{for \, all}~x,y \in R. \end{aligned}$$ Then R is commutative.

Orthogonal symmetric bi-(\sigma,\tau)-derivations in semiprime rings

Orthogonal symmetric bi-(\sigma,\tau)-derivations in semiprime rings

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! R such that F (xy) = F (x)y+xd(y) holds for all x;y2 R. In the present paper we describe the action of generalized derivations satisfying several conditions on Lie ideals of semiprime rings.

A Note on Generalized Jordan Derivations in Semiprime Rings

The main purpose of this paper is to study and investigate some results concerning generalized Jordan derivation and generalized derivation on semiprime ring R, where D an additive mapping on R such that for all and D acts as left centralizer.

Identities with generalized derivations in semiprime rings

AbstractLetRbe a semiprime ring. An additive mappingF:R→Ris called a generalized derivation ofRif there exists a derivationd:R→Rsuch thatF(xy) =F(x)y+xd(y) holds, for allx,y∈R. The objective of the present paper is to study the following situations: (1) [d(x),F(y)] = ±[x,y]; (2) [d(x),F(y)] = ±xοy; (3) [d(x),F(y)] = 0; (4)d(x)οF(y) = ±xοy; (5)d(x)οF(y) = ±[x,y]; (6)d(x)οF(y) = 0; (7)d(x)F(y)±xy∈Z(R), for allx,yin some appropriate subset ofR.

A Note on Constants of Integral Derivations in Semiprime Rings

Let R be a semiprime ring with extended centroid C and with Q its Martindale symmetric ring of quotients. Suppose that δ: R → R is a C-integral derivation. For a subring A of Q, let A(δ) denote the subring of constants of δ in A. We prove that R(δ) and Q(δ) satisfy the same polynomial identities with coefficients in C. In particular, R(δ) is not nil of bounded index.

On Generalized ()-Derivations in Semiprime Rings

Let be a semiprime ring, a nonzero ideal of , and , two epimorphisms of . An additive mapping is generalized -derivation on if there exists a -derivation such that holds for all . In this paper, it is shown that if , then contains a nonzero central ideal of , if one of the following holds: (i) ; (ii) ; (iii) ; (iv) ; (v) for all .

On Skew Derivations in Semiprime Rings

We investigate the commutativity in a (semi-)prime ring R which admits skew derivations δ 1, δ 2 satisfying [δ 1(x), δ 2(y)] = [x, y] for all x, y in a nonzero right ideal of R. This result is a natural generalization of Bell and Daif’s theorem on strong commutativity preserving derivations and a recent result by Ali and Huang.

On functional equations related to derivations in semiprime rings and standard operator algebras

On functional equations related to derivations in semiprime rings and standard operator algebras