Higher derivations and Posner’s second theorem for semiprime rings

Abstract

Let \((d_i)_{i\in \mathbb {N}}\) and \((\delta _j)_{j\in \mathbb {N}}\) be two higher derivations of order m and n respectively on semiprime ring R such that \(d_m(x)x-x\delta _n(x)\in Z(R)\) for all \(x\in I\), where I be an ideal of R. In this article we generalize Posner’s second theorem for higher derivation and prove that either R is commutative or some linear combination of \((\delta _j)\) sends Z(R) to zero.

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Acknowledgements

The author is highly thankful to the referee(s) for valuable suggestions and comments.

Funding

Funding was provided by Science and Engineering Research Board (Grant No. EMR/2016/001550).

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Correspondence to Balchand Prajapati.

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Prajapati, B. Higher derivations and Posner’s second theorem for semiprime rings. Ann Univ Ferrara (2020). https://doi.org/10.1007/s11565-020-00352-4

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Keywords

  • Semiprime ring
  • Extended centroid
  • Higher derivation

Mathematics Subject Classification

  • 16W25
  • 16N60
  • 16R50