Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On Skew Centralizing Traces of Permuting n-Additive Mappings

  • Received : 2013.08.14
  • Accepted : 2013.12.13
  • Published : 2015.03.23
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Abstract

Let R be a ring and $D:R^n{\longrightarrow}R$ be n-additive mapping. A map $d:R{\longrightarrow}R$ is said to be the trace of D if $d(x)=D(x,x,{\ldots}x)$ for all $x{\in}R$. Suppose that ${\alpha},{\beta}$ are endomorphisms of R. For any $a,b{\in}R$, let < a, b > $_{({\alpha},{\beta})}=a{\alpha}(b)+{\beta}(b)a$. In the present paper under certain suitable torsion restrictions it is shown that D = 0 if R satisfies either < d(x), $x^m$ > $_{({\alpha},{\beta})}=0$, for all $x{\in}R$ or ${\ll}$ d(x), x > $_{({\alpha},{\beta})}$, $x^m$ > $_{({\alpha},{\beta})}=0$, for all $x{\in}R$. Further, if < d(x), x > ${\in}Z(R)$, the center of R, for all $x{\in}R$ or < d(x)x - xd(x), x >= 0, for all $x{\in}R$, then it is proved that d is commuting on R. Some more related results are also obtained for additive mapping on R.

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References

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