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Generalized skew-derivations annihilating and centralizing on multilinear polynomials in prime rings

  • Priyadwip Das
  • Basudeb DharaEmail author
  • Sukhendu Kar
Article

Abstract

Let R be a prime ring of characteristic \(\ne 2\), \(Q_r\) its right Martindale quotient ring, C its extended centroid, \(F\ne 0\) a generalized skew derivation of R, \(f(x_1,\ldots ,x_n)\) a multilinear polynomial over C not central-valued on R and S the set of all evaluations of \(f(x_1,\ldots ,x_n)\) in R. If \(a[F(x),x]\in C\) for all \(x\in S\), then there exist \(\lambda \in C\) and \(b\in Q_r\) such that \(F(x)=bx+xb+\lambda x\), for all \(x\in R\) and one of the following holds:
  1. (1)

    \(b\in C\);

     
  2. (2)

    \(f(x_1,\ldots ,x_n)^2 \) is central-valued on R;

     
  3. (3)

    R satisfies \(s_4\), the standered identity of degree 4.

     

Keywords

Prime ring derivation generalized derivation generalized skew derivation extended centroid 

2010 Mathematics Subject Classification

16N60 16W25 16R50 

Notes

Acknowledgements

The authors would like to thank the referee for providing very helpful comments and suggestions. The authors are also grateful to the referee for providing a short proof of Lemma 3.6. The second author is supported by a grant from Science and Engineering Research Board (SERB), DST, New Delhi, India (Grant No. EMR/2016/004043 dated 29-Nov-2016).

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Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of MathematicsJadavpur UniversityKolkataIndia
  2. 2.Department of MathematicsBelda CollegeBeldaIndia

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