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>

On Functional Identities in Prime Rings with Involution II*

2000 ◽  
Vol 28 (7) ◽  
pp. 3169-3183 ◽  
Author(s):  
K.I. Beidar ◽  
M. Brešar ◽  
M A. Chebotar ◽  
W S. Martindale
Keyword(s):  
Prime Rings ◽  
Functional Identities ◽  
Rings With Involution

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.

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.

On generalized derivations and commutativity of prime rings with involution

2017 ◽  
Vol 11 ◽  
pp. 291-300
Author(s):  
Shakir Ali ◽  
Husain Alhazmi
Keyword(s):  
Generalized Derivations ◽  
Prime Rings ◽  
Rings With Involution

Higher-order commutators with power central values on rings and algebras involving generalized derivations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Shakir Ali ◽  
Husain Alhazmi ◽  
Abdul Nadim Khan ◽  
Mohd Arif Raza
Keyword(s):  
Quotient Ring ◽  
Generalized Derivation ◽  
Higher Order ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Left Multiplication ◽  
Central Values ◽  
Power Central ◽  
Rings And Algebras

AbstractLet {\mathfrak{R}} be a ring with center {Z(\mathfrak{R})}. In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation {\mathfrak{F}} such that {([\mathfrak{F}(s^{m})s^{n}+s^{n}\mathfrak{F}(s^{m}),s^{r}]_{k})^{l}\in Z(% \mathfrak{R})} for every {s\in\mathfrak{R}}, then either {\mathfrak{F}(s)=ps} for every {s\in\mathfrak{R}} or {\mathfrak{R}} satisfies {s_{4}} and {\mathfrak{F}(s)=sp} for every {s\in\mathfrak{R}} and {p\in\mathfrak{U}}, the Utumi quotient ring of {\mathfrak{R}}. As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map.

scholarly journals Identities related to generalized derivations and jordan (*,*)-derivations

Filomat ◽  
2021 ◽  
Vol 35 (7) ◽  
pp. 2349-2360
Author(s):  
Amin Hosseinia
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Additive Mapping ◽  
Jordan Derivation ◽  
Generalized Derivations ◽  
Free Ring ◽  
Functional Identities ◽  
Invertible Elements ◽  
Semiprime Algebra

The main purpose of this research is to characterize generalized (left) derivations and Jordan (*,*)-derivations on Banach algebras and rings using some functional identities. Let A be a unital semiprime Banach algebra and let F,G : A ? A be linear mappings satisfying F(x) =-x2G(x-1) for all x ? Inv(A), where Inv(A) denotes the set of all invertible elements of A. Then both F and G are generalized derivations on A. Another result in this regard is as follows. Let A be a unital semiprime algebra and let n > 1 be an integer. Let f,g : A ? A be linear mappings satisfying f (an) = nan-1g(a) = ng(a)an-1 for all a ? A. If g(e) ? Z(A), then f and g are generalized derivations associated with the same derivation on A. In particular, if A is a unital semisimple Banach algebra, then both f and 1 are continuous linear mappings. Moreover, we define a (*,*)-ring and a Jordan (*,*)-derivation. A characterization of Jordan (*,*)-derivations is presented as follows. Let R be an n!-torsion free (*,*)-ring, let n > 1 be an integer and let d : R ? R be an additive mapping satisfying d(an) = ?nj =1 a*n-jd(a)a* j-1 for all a ? R. Then d is a Jordan (*,*)-derivation on R. Some other functional identities are also investigated.

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