Derivations and generalized derivations

Let R be a prime ring of characteristic dierent from 2, L a non-central Lie ideal of R, and m; n xed positive integers. If R admits a generalized derivation F associated with a deviation d such that 􀀀 F(u)2m 􀀀 (F(u))2n 2 Z(R) for all u 2 L, then R satises s4, the standard identity in four variables. Moreover, we also examine the case when R is semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.

In this note we introduce a family of linear operators Dk that contain a sequence of integrals expressions more general in form than the Lanczos Derivative (LD), and show that they all lead to the same limit. The standard LD is a particular case of this family, whose members are labeled by the value of a positive odd integer. The theory is applied to several special cases where the functions are not differentiable in the standard sense, and it is shown that the operators are well-behaved. In addition, we present another couple of operators D , D† -also generalizations of LD-, that are gifted with parallel if not better properties. This fact allows us to claim that the structure of the ordinary LD can be extended in a natural manner by a more generic operator L , which includes a general analytic odd function g(t) in the integrand. In the last part of the work we discuss a mechanism opposed to the Lanzcos method, namely, integration by differentiation.

Consider a ring $R$, which is semiprime and also having $k$-torsion freeness. If $F, d : R\to R$ are two additive maps fulfilling the algebraic identity $$F(x^{n+m})=F(x^m) x^n+ x^m d(x^n)$$ for each $x$ in $R.$ Then $F$ will be a generalized derivation having $d$ as an associated derivation on $R$. On the other hand, in this article, it is also derived that $f$ is a generalized left derivation having a linked left derivation $\delta$ on $R$ if they satisfy the algebraic identity $$f(x^{n+m})=x^n f(x^m)+ x^m \delta(x^n)$$ for each $x$ in $R$ and $k\in \{2, m, n, (n+m-1)!\}$ and at last an application on Banach algebra is presented.

Let R be a semiprime ring, (α1,α2) be automorphisms on R and Δ, δ be additive maps from R to R. If Δ and δ satisfying any one of the following identities:i) Δ(xnyn)=Δ(xn)α1(yn)+α2(xn)δ(yn)ii) Δ(xn)=Δ(xn-m)α1(xm)+α2(xn-m)δ(xm), for all x,y∈Rthen Δ will be generalized (α1,α2)-derivation with associated (α1,α2)-derivation on R.

The present paper is to prove new commutativity theorems for rings (see Theorems 2.1, and 3.1). In addition, applications of commutativity theorems for rings to near-rings, we investigate some polynomial identities (P4) and (P5) with exponents depending on x and y for a certain class of near-rings and then, under some constraints, it is shown that D-near-ring as well as distributively generated (d g) near-rings are commutative. Finally, we close our discussion with some open problems.

1 Centre for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya University, Multan, Pakistan faisalali@bzu.edu.pk, chaudhry@bzu.edu.pk * Corresponding author. ... Let R be a 2-torsion free non-commutative ring with centre Z(R) = {0}. Let F be a ...

Let R be a 2-torsion free σ-prime ring with involution σ, I a nonzero σ-ideal of R and d : R → R a nonzero derivation commuting with σ. In this paper, we first establish that R is commutative if the following conditions: (i) d(x) o x = 0 ∀ x ∈ R, (ii) d(x) o d(y) = 0 ∀ x, y ∈ I, and (iii) d(x o y) = 0 ∀ x, y ∈ I are satisfied. Moreover, also we prove that r is in Z(R) if r in Saσ(R) satisfies d(x) o r = 0 ∀ x in I. Mathematics Subject Classification: 16W10, 16W25, 16U80

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) = xF(y) + d(x)y holds for all x,y ∈ R. It is called a generalized (α,β)−derivation on R if there exists an (α,β)−derivation d: R → R such that the equation F(xy) = F(x)α(y)+β(x)d(y) holds for all x,y ∈ R. In the present paper, we investigate commutativity of a prime ring R, which satisfies certain differential identities on left ideals of R. Moreover some results on commutativity of rings with involutions that satisfy certain identities are proved.

Let R , R' be two prime rings and  n , n be two higher homomorphisms of a ring R for all n  N , in the present paper we show that under certain conditions of R, every Jordan (,)higher homomorphism of a ring R into a prime ring R' is either (,)-higher homomorphism or (,)-higher anti homomorphism.

Let R be a σ-prime ring with characteristic not 2, Z(R) be the center of R, I be a nonzero σ-ideal of R, α, β : R→ R be two automorphisms, d be a nonzero (α, β)-derivation of R and h be a nonzero derivation of R. In the present paper, it is shown that (i) If d (I) ⊂ Cα,β and β commutes with σ then R is commutative. (ii) Let α and β commute with σ. If a ∈ I ∩ Sσ (R) and [d(I), a]α,β ⊂ Cα,β then a ∈ Z(R). (iii) Let α, β and h commute with σ. If dh (I) ⊂ Cα,β and h (I) ⊂ I then R is commutative.

LetRbe a prime ring of characteristic not2,Ua nonzero ideal ofRand0≠da(α,β)-derivation ofRwhereαandβare automorphisms ofR. i)[d(U),a]=0thena∈Zii) Fora,b∈R, the following conditions are equivalent (I)α(a)d(x)=d(x)β(b), for allx∈U(II) Eitherα(a)=β(b)∈CR(d(U))orCR(a)=CR(b)=R′anda[a,x]=[a,x]b(ora[b,x]=[b,x]b) for allx∈U. LetRbe a2-torsion free semiprime ring andUbe a nonzero ideal ofRiii) Letdbe a(α,β)-derivation ofRandgbe a(γ,δ)-derivation ofR. Suppose thatdgis a(αγ,βδ)-derivation andgcommutes bothγandδtheng(x)Uα−1d(y)=0, for allx,y∈Uiv) LetAnn(U)=0anddbe an(α,β)-derivation ofRandgbe a(λ,δ)-derivation ofRsuch thatgcommutes bothγ, andδ. If for allx,y∈U,β−1(d(x))Ug(y)=0=g(x)Uα−1(d(y))thendgis a(αγ,βδ)-derivation onR.

Let N be a prime left near-ring with multiplicative centerZ; and D be a (α, γ)derivation such that δD = Dδ and ΓD = DΓ(i)If D(N)⊂ Z; or [D(N);D(N)] = 0 or [D(N);D(N)]σ, γ= 0; then (N; +)is abelian. (ii) If N is 2-torsion free, d1 is a (α, γ)-derivation and d2 is a derivation on N such that d1d2(N) = 0, then d1 = 0 or d2 = 0.</p

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The purpose of the present paper is to prove some results concerning symmetric generalized biderivations on prime and semiprime rings which partially extend some results of Vukman \cite {V}. Infact we prove that: let $R$ be a prime ring of characteristic not two and $I$ be a nonzro ideal of $R$. If $\Delta$ is a symmetric generalized biderivation on $R$ with associated biderivation $D$ such that $[\Delta(x,x), \Delta(y,y)]=0$ for all $x,y \in I$, then one of the following conditions hold\\ \begin{enumerate} \item $R$ is commutative. \item $\Delta$ acts as a left bimultiplier on $R$. \end{enumerate}

There has been considerable interest in the connection between the structure and the σ-structure of a ring, where σ denotes an involution on a ring. In this context, Oukhtite and Salhi (2006) introduced a new class or we can say an extension of prime rings in the form of σ-prime ring and proved several well-known theorems of prime rings for σ-prime rings. A continuous approach in the direction of σ-prime rings is still on. In this paper, we establish some results for σ-prime rings satisfying certain identities involving generalized derivations on σ-ideals. Finally, we give an example showing that the restrictions imposed on the hypothesis of the various theorems were not superfluous.

A mapping G: R→ R (not necessarily additive) is called multiplicative right αcentralizer if T(xy) = α(x)T(y) for all x, y ∈ R. A mapping G: R → R (not necessarily additive) is called multiplicative (generalized)-(α, β)-reverse derivation if there exists a map (neither necessarily additive or derivation) g : R→ R such that G(xy) = G(y)α(x) + β(y)g(x) for all x,y ∈ R, where α and β are automorphisms on R. The main purpose of this paper is to study some algebraic identities with multiplicative (generalized)-(α, β)-reverse derivations and multiplicative right α-centralizer on the left ideal of a prime ring R. The main objective of the present paper is to investigate the following algebraic identities: (i) G(xy) ± T (x)T(y) = 0 (ii) G(xy) ± T (xy) = 0 (iii) G(xy) ± T(xy) ∈ Z(R) (iv) G(xy) ± G(x)T(y) ∈ Z(R) and (v) G(xy) ± T(x)G(y) = 0 for all x, y in an appropriate subset of R.

‎Let $R$ be a $*$-prime ring with center‎ ‎$Z(R)$‎, ‎$d$ a non-zero $(sigma,tau)$-derivation of $R$ with associated‎ ‎automorphisms $sigma$ and $tau$ of $R$‎, ‎such that $sigma$‎, ‎$tau$‎ ‎and $d$ commute with $'*'$‎. ‎Suppose that $U$ is an ideal of $R$ such that $U^*=U$‎, ‎and $C_{sigma,tau}={cin‎ ‎R~|~csigma(x)=tau(x)c~mbox{for~all}~xin R}.$ In the present paper‎, ‎it is shown that if characteristic of $R$ is different from two and‎ ‎$[d(U),d(U)]_{sigma,tau}={0},$ then $R$ is commutative‎. ‎Commutativity of $R$ has also been established in case if‎ ‎$[d(R),d(R)]_{sigma,tau}subseteq C_{sigma,tau}.$