$k$-power centralizing and $k$-power skew-centralizing maps on‎ ‎triangular rings

Document Type: Research Paper

Author

Department of Mathematics‎, ‎Shanxi University‎, ‎Taiyuan 030006‎, ‎P‎. ‎R‎. ‎China.

Abstract

‎Let $\mathcal A$ and $\mathcal B$ be unital rings‎, ‎and $\mathcal M$‎ ‎be an $(\mathcal A‎, ‎\mathcal B)$-bimodule‎, ‎which is faithful as a‎ ‎left $\mathcal A$-module and also as a right $\mathcal B$-module‎.  ‎Let ${\mathcal U}=Tri(\mathcal A‎, ‎\mathcal M‎, ‎\mathcal‎ ‎B)$ be the triangular ring and ${\mathcal Z}({\mathcal U})$ its‎  ‎center‎. ‎Assume that $f:{\mathcal U}\rightarrow{\mathcal U}$ is a map‎  ‎satisfying $f(x+y)-f(x)-f(y)\in{\mathcal Z}({\mathcal U})$ for all‎ ‎$x,\ y\in{\mathcal U}$ and $k$ is a positive integer‎. ‎It is shown‎  ‎that‎, ‎under some mild conditions‎, ‎the following statements are‎  ‎equivalent‎: ‎(1) $[f(x),x^k]\in{\mathcal Z}({\mathcal U})$ for all‎  ‎$x\in{\mathcal U}$; (2)  $[f(x),x^k]=0$ for all $x\in{\mathcal U}$;‎ ‎(3) $[f(x),x]=0$ for all $x\in{\mathcal U}$; (4) there exist a‎  ‎central element $z\in{\mathcal Z}({\mathcal U})$ and an additive‎ ‎modulo ${\mathcal Z}({\mathcal U})$ map $h:{\mathcal‎
‎U}\rightarrow{\mathcal Z}({\mathcal U})$ such that $f(x)=zx+h(x)$‎ ‎for all $x\in{\mathcal U}$‎. ‎It is also shown that there is no‎ ‎nonzero additive $k$-skew-centralizing maps on triangular rings.
 

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