Rocky Mountain Journal of Mathematics

Strong commutativity preserving maps on rings

Zhaofang Bai and Shuanping Du

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Abstract

Suppose $\R$ is a unital ring having an idempotent element $e$ which satisfies $a{\R}e=0$ implies $a=0$ and $a{\R}(1-e)=0$ implies $a=0$. In this paper, we aim to characterize the map $f:{\R}\rightarrow {\R}$, $f$ is surjective and $[f(x),f(y)]=[x,y]$ for all $x,y\in {\R}$. It is shown that $f(x)=\alpha x +\xi (x)$ for all $x\in {\R}$, where $\alpha \in {\Z}({\R})$, $\alpha^2=1$, and $\xi$ is a map from ${\R}$ into ${\Z}({\R})$. As an application, a characterization of nonlinear surjective maps preserving strong commutativity on von Neumann algebras with no central summands of type $I_1$ is obtained.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 3 (2014), 733-742.

Dates
First available in Project Euclid: 28 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1411945661

Digital Object Identifier
doi:10.1216/RMJ-2014-44-3-733

Mathematical Reviews number (MathSciNet)
MR3264479

Zentralblatt MATH identifier
1312.16041

Subjects
Primary: 16N60. 16U80: Generalizations of commutativity

Keywords
Strong commutativity preserving ring von Neumann algebra

Citation

Bai, Zhaofang; Du, Shuanping. Strong commutativity preserving maps on rings. Rocky Mountain J. Math. 44 (2014), no. 3, 733--742. doi:10.1216/RMJ-2014-44-3-733. https://projecteuclid.org/euclid.rmjm/1411945661


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