On Lie ideals with generalized \((\alpha , \alpha )\)-derivations in prime rings

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Abstract

Let R be a prime ring and \(\alpha \) an automorphism on R. An additive mapping F on R is said to be a generalized \((\alpha , \alpha )\)-derivation on R if there exists an \((\alpha , \alpha )\)-derivation d on R such that \(F(xy)=F(x)\alpha (y)+\alpha (x)d(y)\) holds for all \(x, y\in R\). In this paper our main objective is to study the following identities: (i) \(G(xy) \pm F(x)F(y)\in Z(R);\) (ii) \(G(xy)\pm F(x)F(y)\pm \alpha (yx) =0;\) (iii) \(G(xy) \pm F(x)F(y)\pm \alpha (xy)\in Z(R);\) (iv) \(G(xy)\pm F(x)F(y)\pm \alpha ([x, y]) =0;\) (v) \(G(xy)\pm F(x)F(y)\pm \alpha (x\circ y)=0;\) for all xy in some suitable subset of R, where G and F are two generalized \((\alpha ,\alpha )\)-derivations on R

Keywords

Prime ring Generalized \((\alpha , \alpha )\)-derivation Automorphism Square closed Lie ideal 

Mathematics Subject Classification

16W25 16N60 16U80 

Notes

Acknowledgements

The author are greatly indebted to the referee for his/her several useful suggestions and valuable comments.

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© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology DelhiHauz Khas, New DelhiIndia

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