Nonadditive strong commutativity preserving derivations and endomorphisms
Bull. Korean Math. Soc. 2014 Vol. 51, No. 4, 1127-1133
https://doi.org/10.4134/BKMS.2014.51.4.1127
Published online July 31, 2014
Wei Zhang and Xiaowei Xu
Jilin University, Jilin University
Abstract : Let $S$ be a nonempty subset of a ring $R$. A map $f:R\rightarrow R$ is called strong commutativity preserving on $S$ if $[f(x),f(y)]=[x,y]$ for all $x,y\in S$, where the symbol $[x,y]$ denotes $xy-yx$. Bell and Daif proved that if a derivation $D$ of a semiprime ring $R$ is strong commutativity preserving on a nonzero right ideal $\rho$ of $R$, then $\rho\subseteq Z$, the center of $R$. Also they proved that if an endomorphism $T$ of a semiprime ring $R$ is strong commutativity preserving on a nonzero two-sided ideal $I$ of $R$ and not identity on the ideal $I\cup T^{-1}(I)$, then $R$ contains a nonzero central ideal. This short note shows that the conclusions of Bell and Daif are also true without the additivity of the derivation $D$ and the endomorphism $T$.
Keywords : semiprime ring, prime ring, strong commutativity preserving map
MSC numbers : 16W25, 16N60
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