Identities related to generalized derivation on ideal in prime rings

2015 ◽  
Vol 57 (4) ◽  
pp. 809-821 ◽  
Author(s):  
Shailesh Kumar Tiwari ◽  
Rajendra K. Sharma ◽  
Basudeb Dhara
Keyword(s):  
Generalized Derivation ◽  
Prime Rings

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.

Derivations vanishing on commutators with generalized derivation of order 2 in prime rings

2016 ◽  
Vol 45 (8) ◽  
pp. 3542-3554 ◽  
Author(s):  
S. K. Tiwari ◽  
R. K. Sharma ◽  
B. Dhara
Keyword(s):  
Generalized Derivation ◽  
Prime Rings

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.

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 Generalized Derivations of Prime Rings

2007 ◽  
Vol 2007 ◽  
pp. 1-6 ◽  
Author(s):  
Huang Shuliang
Keyword(s):  
Prime Ring ◽  
Additive Function ◽  
Generalized Derivation ◽  
Lie Ideal ◽  
Generalized Derivations ◽  
Prime Rings

LetRbe an associative prime ring,Ua Lie ideal such thatu2∈Ufor allu∈U. An additive functionF:R→Ris called a generalized derivation if there exists a derivationd:R→Rsuch thatF(xy)=F(x)y+xd(y)holds for allx,y∈R. In this paper, we prove thatd=0orU⊆Z(R)if any one of the following conditions holds: (1)d(x)∘F(y)=0, (2)[d(x),F(y)=0], (3) eitherd(x)∘F(y)=x∘yord(x)∘F(y)+x∘y=0, (4) eitherd(x)∘F(y)=[x,y]ord(x)∘F(y)+[x,y]=0, (5) eitherd(x)∘F(y)−xy∈Z(R)ord(x)∘F(y)+xy∈Z(R), (6) either[d(x),F(y)]=[x,y]or[d(x),F(y)]+[x,y]=0, (7) either[d(x),F(y)]=x∘yor[d(x),F(y)]+x∘y=0for allx,y∈U.

Derivations vanishing on commutator identity involving generalized derivation on multilinear polynomials in prime rings

2018 ◽  
Vol 47 (2) ◽  
pp. 800-813
Author(s):  
Mohammad Ashraf ◽  
Vincenzo De Filippis ◽  
Sajad Ahmad Pary ◽  
Shailesh Kumar Tiwari
Keyword(s):  
Generalized Derivation ◽  
Prime Rings ◽  
Commutator Identity ◽  
Multilinear Polynomials

Local Generalized Derivations in Prime Rings with Idempotents

2010 ◽  
Vol 17 (02) ◽  
pp. 295-300 ◽  
Author(s):  
Yu Wang
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Additive Map

Let R be a prime ring with a nontrivial idempotent. In this paper, we prove that if g is an additive map of R into itself such that xg(y)z = 0 for all x, y, z ∈ R with xy = yz = 0, then g is a generalized derivation. As an application of this result, we show that every local generalized derivation in a prime ring with a nontrivial idempotent is a generalized derivation.

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 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.

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