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.

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 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>

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.

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 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.

scholarly journals A Characterization of Derivations in Prime Rings with Involution

2019 ◽  
Vol 12 (3) ◽  
pp. 1138-1148
Author(s):  
Shakir Ali ◽  
M. Rahman Mozumder ◽  
Adnan Abbasi ◽  
M. Salahuddin Khan
Keyword(s):  
Noncommutative Ring ◽  
Prime Rings ◽  
Differential Identities ◽  
Rings With Involution ◽  
Torsion Free

The purpose of this paper is to investigate $*$-differential identities satisfied by pair of derivations on prime rings with involution. In particular, we prove that if a 2-torsion free noncommutative ring $R$ admit nonzero derivations $d_1, d_2$ such that $[d_1(x), d_2(x^*)]=0$ for all $x\in R$, then $d_1=\lambda d_2$, where $\lambda\in C$. Finally, we provide an example to show that the condition imposed in the hypothesis of our results are necessary.

On certain classes of generalized derivations

2019 ◽  
Vol 69 (5) ◽  
pp. 1023-1032 ◽  
Author(s):  
Omar Ait Zemzami ◽  
Lahcen Oukhtite ◽  
Shakir Ali ◽  
Najat Muthana
Keyword(s):  
Generalized Derivations ◽  
Prime Rings ◽  
Rings With Involution

Abstract Our purpose in this paper is to investigate some particular classes of generalized derivations and their relationship with commutativity of prime rings with involution. Some well-known results characterizing commutativity of prime rings have been generalized. Furthermore, we provide examples to show that the assumed restrictions cannot be relaxed.

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