A Sivakameshwara Kumar’s scientific contributions

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Publications (7)


Left Generalized Derivations on Prime gamma Rings
  • Article
  • Full-text available

January 2018

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195 Reads

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1 Citation

Annals of Pure and Applied Mathematics

C Jaya

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K Nagesh

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A Sivakameshwara Kumar

Let be a prime-ring with 2-torsion free, a nonzero ideal of M and : → a left generalized derivation of , with associated nonzero derivation d on. If () ∈ () for all ∈ , then is a commutative-ring.

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Lie ideal and generalized Jordan reverse derivations on semiprime rings

December 2017

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184 Reads

Far East Journal of Mathematical Sciences (FJMS)

Let R be a 2-torsion free semiprime ring in which x² = 0 implies x = 0. Let g: R → R be a generalized Jordan reverse derivation associated with the Jordan reverse derivation d : R → R. Then g is a generalized reverse derivation on R. Thus, there exists q ∈ Qr (S). The Martindale quotient ring of S, such that g(x) = qx + d(x), for all x ∈ R.





Citations (2)


... In [10] , El-Soufi and Aboubakr proved that J ⊆ Z(R) under specific properties, where R is a 2-torsion free prime ring with center Z(R) admitting a generalized derivation F associated with a derivation d , J is a nonzero Jordan ideal. In addition, Ibraheem in [11] showed that if f is a generalized reverse derivation on R such that f is commuting and centralizing on a right ideal I of R, then R is a commutative, where R is a prime ring and d is a reverse derivation on R. Moreover, in [1], Abu Nawas and Al-Omary investigated the commutativity of R such that R is a * -prime ring admitting generalized (α, β)-derivations F and G associated with (α, β)−derivations d and g, respectively, that satisfying certain properties. ...

Reference:

On Ideals and Commutativity of Prime Rings with Generalized Derivations
Lie Ideals and Jordan Generalized Reverse Derivations of Prime Rings

... To prove that R is Г-regular Let (α ̶ α ω α) ∈ J and there is a unit ω ∈ R * and where α ∈ R such that α ̶ α ω α = (α ̶ α ω α)ߛ(α ̶ α ω α) where ߛ ∈ I * . Then α = α ̶ α ω α + α ω α= (α ̶ α ω α) ߛ (α ̶ α ω α) + α ω α = (α γ⎼ α ω α γ)(α ⎼ α ω α)+ α ω α= α γ α ⎼α γ α ω α ⎼α ω α γ α + α ω α γ α ω α+ α ω α = α(ߛ ̶ ߛ α ω ̶ ω α ߛ ̶ + ω α ߛ α ω + ω)α=α ߜ α, Where ߜ = γ ̶ ߛ α ω ⎼ω α ߛ ̶ + ω α ߛ α ω + ω ∈ R * [11]. Since I * ⊆R * and R * is an ideal in (R, Г).Hence R is Гregular. ...

Left Generalized Derivations on Prime gamma Rings

Annals of Pure and Applied Mathematics