Generalized Derivations Acting as Homomorphisms and Anti-Homomorphisms of Gamma Rings

Dey, Kalyan Kumar; Paul, Akhil Chandra

doi:10.3329/jsr.v4i1.7911

Cited by 3 publications

(2 citation statements)

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“…In Γ-rings, Dey and Paul [9] proved that if D is a generalized derivation of a prime Γ-ring M with an associated derivation In this article, the above mentioned results following [11] in classical rings are extended to those in gamma rings with derivation acting as a homomorphism and as an anti-homomorphism on σ-prime Γ-rings. Our objective is to prove that…”

Section: Introductionmentioning

confidence: 95%

Derivations Acting as Homomorphisms and as Anti-homomorphisms in σ-Lie Ideals of σ-Prime Gamma Rings

Paul¹,

Chakraborty²

2015

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Let U be a non-zero σ-square closed Lie ideal of a 2-torsion free σ-prime Γ-ring M satisfying the condition aαbβc = aβbαc for all a, b, c ∈ M and α, β ∈ Γ, and let d be a derivation of M such that dσ = σd. We prove here that (i) if d acts as a homomorphism on U , then d = 0 or U ⊆ Z(M), where Z(M) is the centre of M ; and (ii) if d acts as an anti-homomorphism on U , then d = 0 or U ⊆ Z(M).

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Section: Introductionmentioning

confidence: 95%

Derivations Acting as Homomorphisms and as Anti-homomorphisms in σ-Lie Ideals of σ-Prime Gamma Rings

Paul¹,

Chakraborty²

2015

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“…in [6]. In Γ-rings, Dey and Paul [4] proved that if D is a generalized derivation of a prime Γ-ring M with an associated derivation d of M which acts as a homomorphism and an anti-homomorphism on a non-zero ideal I of M, then d = 0 or M is commutative. Afterwards, Chakraborty and Paul [11] worked on kderivation of a semiprime Γ-ring in the sense of Nobusawa [10] and proved that d = 0 where d is a k-derivation acting as a k-endomorphism and as an anti-kendomorphism, the above mentioned results following [6] in classical rings are extended to those in gamma rings with derivation acting as a homomorphism and as an anti-homomorphism on σ-prime Γ-rings.…”

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confidence: 99%

$\Gamma^*$-DERIVATION ACTING AS AN ENDOMORPHISM AND AS AN ANTI-ENDOMORPHISM IN SEMIPRIME $\Gamma$-RING M WITH INVOLUTION

Kadhim¹,

Sulaiman²,

Majeed³

2015

Int. J. of Pure and Appl. Math.

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The commutativity of prime Γ-rings with generalized skew derivations

Huang

Ali

2016

Georgian Mathematical Journal

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Let M be a prime Γ-ring with center {Z(M)}, and let θ be an automorphism of M. An additive map {d:M\to M} is called a skew derivation if {d(x\alpha y)=d(x)\alpha y+\theta(x)\alpha d(y)} for all {x,y\in M}, {\alpha\in\Gamma}. An additive map {F:M\to M} is called a generalized skew derivation if there exists a skew derivation {d:M\to M} such that {F(x\alpha y)=F(x)\alpha y+\theta(x)\alpha d(y)} holds for all {x,y\in M}, {\alpha\in\Gamma}. In the present paper, our main objective is to prove some commutativity results for prime Γ-rings M admitting a generalized skew derivation F satisfying anyone of the properties:(i){F(x\alpha y)\pm x\alpha y\in Z(M)},(ii){F(x\alpha y)\pm y\alpha x\in Z(M)},(iii){F(x)\alpha F(y)\pm x\alpha y\in Z(M)},(iv){F([x,y]_{\alpha})\pm[x,y]_{\alpha}=0},(v){F(\langle x,y\rangle_{\alpha})\pm\langle x,y\rangle_{\alpha}=0}for all {x,y\in I} and {\alpha\in\Gamma}. In fact, we obtain rather more general results which unify, extend and complement several well-known results proved in [3, 4, 5, 6, 32].

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Generalized Derivations Acting as Homomorphisms and Anti-Homomorphisms of Gamma Rings

Cited by 3 publications

References 3 publications

Derivations Acting as Homomorphisms and as Anti-homomorphisms in σ-Lie Ideals of σ-Prime Gamma Rings

Derivations Acting as Homomorphisms and as Anti-homomorphisms in σ-Lie Ideals of σ-Prime Gamma Rings

$\Gamma^*$-DERIVATION ACTING AS AN ENDOMORPHISM AND AS AN ANTI-ENDOMORPHISM IN SEMIPRIME $\Gamma$-RING M WITH INVOLUTION

The commutativity of prime Γ-rings with generalized skew derivations

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