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 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.

Engel type identities with generalized derivations in prime rings

2018 ◽  
Vol 11 (04) ◽  
pp. 1850055
Author(s):  
Basudeb Dhara ◽  
Krishna Gopal Pradhan ◽  
Shailesh Kumar Tiwari
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Nonzero Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Fixed Integer ◽  
Derivation Formula ◽  
Ring Of Quotients ◽  
Outer Derivation

Let [Formula: see text] be a noncommutative prime ring with its Utumi ring of quotients [Formula: see text], [Formula: see text] the extended centroid of [Formula: see text], [Formula: see text] a generalized derivation of [Formula: see text] and [Formula: see text] a nonzero ideal of [Formula: see text]. If [Formula: see text] satisfies any one of the following conditions: (i) [Formula: see text], [Formula: see text], [Formula: see text], (ii) [Formula: see text], where [Formula: see text] is a fixed integer, then one of the following holds: (1) there exists [Formula: see text] such that [Formula: see text] for all [Formula: see text]; (2) [Formula: see text] satisfies [Formula: see text] and there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text]; (3) char [Formula: see text], [Formula: see text] satisfies [Formula: see text] and there exist [Formula: see text] and an outer derivation [Formula: see text] of [Formula: see text] such that [Formula: see text] for all [Formula: see text].

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.

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.

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.

scholarly journals Some Theorems for Sigma Prime Rings with Differential Identities on Sigma Ideals

ISRN Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Mohd Rais Khan ◽  
Mohd Mueenul Hasnain
Keyword(s):  
Prime Ring ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Differential Identities ◽  
New Class ◽  
Continuous Approach

There has been considerable interest in the connection between the structure and the σ-structure of a ring, where σ denotes an involution on a ring. In this context, Oukhtite and Salhi (2006) introduced a new class or we can say an extension of prime rings in the form of σ-prime ring and proved several well-known theorems of prime rings for σ-prime rings. A continuous approach in the direction of σ-prime rings is still on. In this paper, we establish some results for σ-prime rings satisfying certain identities involving generalized derivations on σ-ideals. Finally, we give an example showing that the restrictions imposed on the hypothesis of the various theorems were not superfluous.

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 Generalized Derivations Acting as Homomorphisms and Anti-Homomorphisms on Lie Ideals of Prime Rings

2016 ◽  
Vol 35 ◽  
pp. 73-77
Author(s):  
Akhil Chandra Paul ◽  
Sujoy Chakraborty
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Lie Ideal ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Square Closed Lie Ideal ◽  
Torsion Free

Let U be a non-zero square closed Lie ideal of a 2-torsion free prime ring R and f a generalized derivation of R with the associated derivation d of R. If f acts as a homomorphism and as an anti-homomorphism on U, then we prove that d = 0 or U € Z(R), the centre of R.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 73-77

Generalized Derivations on Lie Ideals and Power Values on Prime Rings

2015 ◽  
Vol 65 (5) ◽  
Author(s):  
Giovanni Scudo ◽  
Abu Zaid Ansari
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Generalized Derivation ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Utumi Quotient Ring ◽  
Standard Identity

AbstractLet R be a non-commutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, L a non-central Lie ideal of R, G a non-zero generalized derivation of R.If [G(u), u](1) R satisfies the standard identity s(2) there exists γ ∈ C such that G(x) = γx for all x ∈ R.

Generalized Derivations in Rings with Involution

2017 ◽  
Vol 24 (03) ◽  
pp. 393-399 ◽  
Author(s):  
Nadeem Ahmad Dar ◽  
Abdul Nadim Khan
Keyword(s):  
Prime Ring ◽  
Related Result ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
Derivation Formula ◽  
Rings With Involution ◽  
Strong Commutativity Preserving ◽  
Strong Commutativity

The main purpose of this paper is to study generalized derivations in rings with involution which behave like strong commutativity preserving mappings. In fact, we prove the following result: Let R be a noncommutative prime ring with involution of the second kind such that char [Formula: see text]. If R admits a generalized derivation [Formula: see text] associated with a derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text], then [Formula: see text] for all [Formula: see text] or [Formula: see text] for all [Formula: see text]. Moreover, a related result is also obtained.

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