Generalized derivations, quasiderivations and centroids of Bihom-Lie triple system

2021 ◽  
pp. 2250111
Author(s):  
Abdelkader Ben Hassine
Keyword(s):  
Triple System ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
Lie Triple System ◽  
Derivation Algebra

In this paper, we give some properties of the generalized derivation algebra [Formula: see text] of a Bihom-Lie triple system [Formula: see text]. In particular, we prove that [Formula: see text], the sum of the quasiderivation algebra and the quasicentroid. We also prove that [Formula: see text] can be embedded as derivations in a larger Bihom-Lie triple system.

scholarly journals Generalized derivations of Lie triple systems

2016 ◽  
Vol 14 (1) ◽  
pp. 260-271 ◽  
Author(s):  
Jia Zhou ◽  
Liangyun Chen ◽  
Yao Ma
Keyword(s):  
Triple System ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
Lie Triple System ◽  
Triple Systems ◽  
Basic Properties ◽  
Derivation Algebra ◽  
The Relationship

AbstractIn this paper, we present some basic properties concerning the derivation algebra Der (T), the quasiderivation algebra QDer (T) and the generalized derivation algebra GDer (T) of a Lie triple system T, with the relationship Der (T) ⊆ QDer (T) ⊆ GDer (T) ⊆ End (T). Furthermore, we completely determine those Lie triple systems T with condition QDer (T) = End (T). We also show that the quasiderivations of T can be embedded as derivations in a larger Lie triple system.

scholarly journals Generalized derivations in prime and semiprime

2015 ◽  
Vol 34 (2) ◽  
pp. 29
Author(s):  
Shuliang Huang ◽  
Nadeem Ur Rehman
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Generalized Derivation ◽  
Nonzero Ideal ◽  
Generalized Derivations ◽  
Positive Integers

Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $m, n$  fixed positive integers.  If $R$ admits a generalized derivation $F$ associated with a  nonzero derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for  all $x,y\in I$, then $R$ is commutative. Moreover  we also examine the case when $R$ is a semiprime ring.

Some results on ∗-differential identities in prime rings

2021 ◽  
Author(s):  
Deepak Kumar ◽  
Bharat Bhushan ◽  
Gurninder S. Sandhu
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Differential Identities ◽  
Derivation Formula

Let [Formula: see text] be a prime ring with involution ∗ of the second kind. An additive mapping [Formula: see text] is called generalized derivation if there exists a unique derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text] In this paper, we investigate the structure of [Formula: see text] and describe the possible forms of generalized derivations of [Formula: see text] that satisfy specific ∗-differential identities. Precisely, we study the following situations: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] for all [Formula: see text] Moreover, we construct some examples showing that the restrictions imposed in the hypotheses of our theorems are not redundant.

On continuous linear generalized derivation in rings and Banach algebras

2019 ◽  
pp. 2150020
Author(s):  
Nadeem ur Rehman
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Generalized Derivation ◽  
Continuous Linear ◽  
Generalized Derivations ◽  
Derivation Formula

In this paper, we investigate the commutativity of a prime Banach algebra [Formula: see text] which admits a nonzero continuous linear generalized derivation [Formula: see text] associated with continuous linear derivation [Formula: see text] such that either [Formula: see text] or [Formula: see text] for intergers [Formula: see text] and [Formula: see text] and sufficiently many [Formula: see text]. Further, similar results are also obtained for unital prime Banach algebra [Formula: see text] which admits a nonzero continuous linear generalized derivations [Formula: see text] satisfying either [Formula: see text] or [Formula: see text] for an integer [Formula: see text] and sufficiently many [Formula: see text].

scholarly journals Some identities involving multiplicative generalized derivations in orime and semiprime rings

2018 ◽  
Vol 36 (1) ◽  
pp. 25 ◽  
Author(s):  
Basudeb Dhara
Keyword(s):  
Generalized Derivation ◽  
Generalized Derivations ◽  
Semiprime Rings

Let $R$ be a ring with center $Z(R)$. A mapping $F:R\rightarrow R$ is called a multiplicative generalized derivation, if $F(xy)=F(x)y+xg(y)$ is fulfilled for all $x,y\in R$, where $g:R\rightarrow R$ is a derivation. In the present paper, our main object is to study the situations: (1) $F(xy)- F(x)F(y)\in Z(R)$, (2) $F(xy)+ F(x)F(y)\in Z(R)$, (3) $F(xy)- F(y)F(x)\in Z(R)$, (4) $F(xy)+ F(y)F(x)\in Z(R)$, (5) $F(xy)- g(y)F(x)\in Z(R)$; for all $x,y$ in some suitable subset of $R$.

On Some Generalized Identities with Derivations on Multilinear Polynomials

2010 ◽  
Vol 17 (02) ◽  
pp. 319-336 ◽  
Author(s):  
Luisa Carini ◽  
Vincenzo De Filippis ◽  
Onofrio Mario Di Vincenzo
Keyword(s):  
Commutative Ring ◽  
Generalized Derivation ◽  
Multilinear Polynomial ◽  
Generalized Derivations ◽  
Multilinear Polynomials

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, Z(R) the center of R, f(x1,…,xn) a non-central multilinear polynomial over K, d and δ derivations of R, a and b fixed elements of R. Denote by f(R) the set of all evaluations of the polynomial f(x1,…,xn) in R. If a[d(u),u] + [δ (u),u]b = 0 for any u ∈ f(R), we prove that one of the following holds: (i) d = δ = 0; (ii) d = 0 and b = 0; (iii) δ = 0 and a = 0; (iv) a, b ∈ Z(R) and ad + bδ = 0. We also examine some consequences of this result related to generalized derivations and we prove that if d is a derivation of R and g a generalized derivation of R such that g([d(u),u]) = 0 for any u ∈ f(R), then either g = 0 or d = 0.

scholarly journals Generalized Derivations in Semiprime Gamma Rings

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Kalyan Kumar Dey ◽  
Akhil Chandra Paul ◽  
Isamiddin S. Rakhimov
Keyword(s):  
Generalized Derivation ◽  
Additive Mapping ◽  
Generalized Derivations ◽  
Gamma Rings ◽  
Torsion Free

LetMbe a 2-torsion-free semiprimeΓ-ring satisfying the conditionaαbβc=aβbαcfor alla,b,c∈M,  α,β∈Γ, and letD:M→Mbe an additive mapping such thatD(xαx)=D(x)αx+xαd(x)for allx∈M,  α∈Γand for some derivationdofM. We prove thatDis a generalized derivation.

scholarly journals Two-Sided Annihilator Condition with Generalized Derivations on Multilinear Polynomials

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
V. De Filippis ◽  
G. Scudo ◽  
L. Sorrenti
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Generalized Derivation ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Utumi Quotient Ring ◽  
Multilinear Polynomials

Let R be a prime ring of characteristic different from 2, with extended centroid C, U its two-sided Utumi quotient ring, F a nonzero generalized derivation of R, f(x1,…,xn) a noncentral multilinear polynomial over C in n noncommuting variables, and a,b∈R such that a[F(f(r1,…,rn)),f(r1,…,rn)]b=0 for any r1,…,rn∈R. Then one of the following holds: (1) a=0; (2) b=0; (3) there exists λ∈C such that F(x)=λx, for all x∈R; (4) there exist q∈U and λ∈C such that F(x)=(q+λ)x+xq, for all x∈R, and f(x1,…,xn)2 is central valued on R; (5) there exist q∈U and λ,μ∈C such that F(x)=(q+λ)x+xq, for all x∈R, and aq=μa, qb=μb.

scholarly journals On Generalized ()-Derivations in Semiprime Rings

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Basudeb Dhara ◽  
Atanu Pattanayak
Keyword(s):  
Semiprime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Nonzero Ideal ◽  
Generalized Derivations ◽  
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 .

The Frattini Subsystem of a Lie Triple System

2009 ◽  
Vol 37 (10) ◽  
pp. 3750-3759 ◽  
Author(s):  
Zhixue Zhang ◽  
Liangyun Chen ◽  
Wenli Liu ◽  
Ximei Bai
Keyword(s):  
Triple System ◽  
Lie Triple System
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