On Skew Centralizing Traces of Permuting n-Additive Mappings

• Journal title : Kyungpook mathematical journal
• Volume 55, Issue 1,  2015, pp.1-12
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2015.55.1.1
Title & Authors
On Skew Centralizing Traces of Permuting n-Additive Mappings

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