Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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NOTES ON SYMMETRIC SKEW n-DERIVATION IN RINGS

  • Koc, Emine (Department of Mathematics Cumhuriyet University) ;
  • Rehman, Nadeem ur (Department of Mathematics Aligarh Muslim University)
  • Received : 2017.11.14
  • Accepted : 2018.04.11
  • Published : 2018.10.31
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Abstract

Let R be a prime ring (or semiprime ring) with center Z(R), I a nonzero ideal of R, T an automorphism of $R,S:R^n{\rightarrow}R$ be a symmetric skew n-derivation associated with the automorphism T and ${\Delta}$ is the trace of S. In this paper, we shall prove that S($x_1,{\ldots},x_n$) = 0 for all $x_1,{\ldots},x_n{\in}R$ if any one of the following holds: i) ${\Delta}(x)=0$, ii) [${\Delta}(x),T(x)]=0$ for all $x{\in}I$. Moreover, we prove that if $[{\Delta}(x),T(x)]{\in}Z(R)$ for all $x{\in}I$, then R is a commutative ring.

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References

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