scholarly journals Generalized Derivations on Prime Near Rings

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Asma Ali ◽  
Howard E. Bell ◽  
Phool Miyan
Keyword(s):  
Generalized Derivation ◽  
Additive Mapping ◽  
Generalized Derivations ◽  
Semigroup Ideal

LetNbe a near ring. An additive mappingf:N→Nis said to be a right generalized (resp., left generalized) derivation with associated derivationdonNiff(xy)=f(x)y+xd(y)(resp.,f(xy)=d(x)y+xf(y)) for allx,y∈N. A mappingf:N→Nis said to be a generalized derivation with associated derivationdonNiffis both a right generalized and a left generalized derivation with associated derivationdonN. The purpose of the present paper is to prove some theorems in the setting of a semigroup ideal of a 3-prime near ring admitting a generalized derivation, thereby extending some known results on derivations.

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.

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 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 .

scholarly journals Semigroup ideals and generalized semiderivations of prime near rings

2018 ◽  
Vol 37 (4) ◽  
pp. 25-45
Author(s):  
Asma Ali ◽  
Abdelkarim Boua ◽  
Farhat Ali
Keyword(s):  
Additive Mapping ◽  
Generalized Derivations ◽  
Semigroup Ideal

Let N be a near ring. An additive mapping F : N 􀀀! N is said to be a generalized semiderivation on N if there exists a semiderivation d : N 􀀀! N associated with a function g : N 􀀀! N such that F(xy) = F(x)y + g(x)d(y) = d(x)g(y) + xF(y) and F(g(x)) = g(F(x)) for all x; y 2 N. In this paper we prove some theorems in the setting of semigroup ideal of a 3-prime near ring admitting a nonzero generalized semiderivation associated with a nonzero semiderivation, thereby extending some known results on derivations, semiderivations and generalized derivations.

scholarly journals Stability and Superstability of Generalized (, )-Derivations in Non-Archimedean Algebras: Fixed Point Theorem via the Additive Cauchy Functional Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
M. Eshaghi Gordji ◽  
M. B. Ghaemi ◽  
G. H. Kim ◽  
Badrkhan Alizadeh
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Generalized Derivations ◽  
Cauchy Functional Equation ◽  
Cauchy’S Functional Equation ◽  
Additive Cauchy Functional Equation

Let be an algebra, and let , be ring automorphisms of . An additive mapping is called a -derivation if for all . Moreover, an additive mapping is said to be a generalized -derivation if there exists a -derivation such that for all . In this paper, we investigate the superstability of generalized -derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchy’s functional equation.

scholarly journals On Ideals and Commutativity of Prime Rings with Generalized Derivations

2018 ◽  
Vol 11 (1) ◽  
pp. 79 ◽  
Author(s):  
Mohammad Khalil Abu Nawas ◽  
Radwan M. Al-Omary
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Differential Identities

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.

scholarly journals A classification of generalized derivations in rings with involution

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1439-1452
Author(s):  
Bharat Bhushan ◽  
Gurninder Sandhu ◽  
Shakir Ali ◽  
Deepak Kumar
Keyword(s):  
Generalized Derivation ◽  
Additive Mapping ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Rings With Involution

Let R be a ring. An additive mapping F : R ? R is called a generalized derivation if there exists a derivation d of R such that F(xy) = F(x)y + xd(y) for all x,y ? R. The main purpose of this paper is to characterize some specific classes of generalized derivations of rings. Precisely, we describe the structure of generalized derivations of noncommutative prime rings with involution that belong to a particular class of generalized derivations. Consequently, some recent results in this line of investigation have been extended. Moreover, some suitable examples showing that the assumed hypotheses are crucial, are also given.

scholarly journals Remarks on Generalized Derivations in Prime and Semiprime Rings

2010 ◽  
Vol 2010 ◽  
pp. 1-6 ◽  
Author(s):  
Basudeb Dhara
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Nonzero Ideal ◽  
Generalized Derivations ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

LetRbe a ring with centerZandIa nonzero ideal ofR. An additive mappingF:R→Ris called a generalized derivation ofRif there exists a derivationd:R→Rsuch thatF(xy)=F(x)y+xd(y)for allx,y∈R. In the present paper, we prove that ifF([x,y])=±[x,y]for allx,y∈IorF(x∘y)=±(x∘y)for allx,y∈I, then the semiprime ringRmust contains a nonzero central ideal, providedd(I)≠0. In caseRis prime ring,Rmust be commutative, providedd≠0. The cases (i)F([x,y])±[x,y]∈Zand (ii)F(x∘y)±(x∘y)∈Zfor allx,y∈Iare also studied.

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.

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].

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