Composition and orthogonality of derivations with multilinear polynomials in prime rings

Abstract

Let R be a non commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C. Let d and \(\delta \) be two derivations of R and S be the set of evaluations of a multilinear polynomial \(f(x_1,\ldots ,x_n)\) over C which is not central valued. Let \(p,q\in R\). We prove the followings.

  1. (1)

    If \(pud\delta (u)+\delta d(u)uq=0\) for all \(u\in S\) and \(p+q\notin C\). Then either \(d=0\) or \(\delta =0\).

  2. (2)

    If \(pud(u)+d(u)uq=0\) for all \(u\in S\). Then either \(d=0\) or \(p=q\in C\), \(d(x)=[a,x]\) for some \(a\in U\) and \(f(x_1,\ldots ,x_n)^2\) is central valued.

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Acknowledgements

The authors are highly thankful to the referee for his/her several useful suggestions. First author is partially supported by the research Grant DST-SERB EMR/2016/001550.

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Correspondence to Balchand Prajapati.

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Prajapati, B., Gupta, C. Composition and orthogonality of derivations with multilinear polynomials in prime rings. Rend. Circ. Mat. Palermo, II. Ser 69, 1279–1294 (2020). https://doi.org/10.1007/s12215-019-00473-6

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Keywords

  • Prime ring
  • Derivation
  • Orthogonal derivations
  • Extended centroid
  • Utumi quotient ring

Mathematics Subject Classification

  • 16W25
  • 16N60