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.

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.

Some commutativity theorems for rings with involution involving generalized derivations

2019 ◽  
Vol 12 (01) ◽  
pp. 1950001 ◽  
Author(s):  
My Abdallah Idrissi ◽  
Lahcen Oukhtite
Keyword(s):  
Generalized Derivation ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Commutativity Theorems ◽  
Rings With Involution

Our purpose in this paper is to investigate commutativity of a ring with involution [Formula: see text] which admits a generalized derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.

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 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 A characterization of $ b- $generalized derivations on prime rings with involution

2021 ◽  
Vol 7 (2) ◽  
pp. 2413-2426
Author(s):  
Mohd Arif Raza ◽  
◽  
Abdul Nadim Khan ◽  
Husain Alhazmi ◽  
Keyword(s):  
Generalized Derivations ◽  
Prime Rings ◽  
Functional Identities ◽  
Rings With Involution

<abstract><p>In this note, we characterize $ b- $generalized derivations which are strong commutative preserving (SCP) on $ \mathscr{R} $. Moreover, we also discuss and characterize $ b- $generalized derivations involving certain $ \ast- $differential/functional identities on rings possessing involution.</p></abstract>

Centralizing and Commuting Left Generalized Derivations on Prime Rings

2015 ◽  
Vol 11 ◽  
pp. 1-3 ◽  
Author(s):  
C. Jaya Subba Reddy ◽  
S. Mallikarjuna Rao ◽  
V. Vijaya Kumar
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
Prime Rings

Let R be a prime ring and d a derivation on R. If is a left generalized derivation on R such that ƒ is centralizing on a left ideal U of R, then R is commutative.

Differential identities on prime rings with involution

2018 ◽  
Vol 17 (09) ◽  
pp. 1850163 ◽  
Author(s):  
A. Mamouni ◽  
B. Nejjar ◽  
L. Oukhtite
Keyword(s):  
Generalized Derivations ◽  
Prime Rings ◽  
Differential Identities ◽  
Rings With Involution

In this paper, we investigate commutativity of prime rings [Formula: see text] with involution ∗ of the second kind in which generalized derivations satisfy certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Furthermore, we provide an example to show that the restriction imposed on the involution is not superfluous.

A study of differential prime rings with involution

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Lahcen Oukhtite ◽  
Omar Ait Zemzami
Keyword(s):  
Generalized Derivations ◽  
Prime Rings ◽  
Rings With Involution

Abstract The main goal of the present paper is to study some results concerning generalized derivations of prime rings with involution. Moreover, we provide examples to show that the assumed restriction cannot be relaxed.

scholarly journals On Maps of Period 2 on Prime and Semiprime Rings

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
H. E. Bell ◽  
M. N. Daif
Keyword(s):  
Generalized Derivations ◽  
Prime Rings ◽  
Period 2 ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Rings With Involution

A mapfof the ringRinto itself is of period 2 iff2x=xfor allx∈R; involutions are much studied examples. We present some commutativity results for semiprime and prime rings with involution, and we study the existence of derivations and generalized derivations of period 2 on prime and semiprime rings.

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