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On Automorphisms and Commutativity in Semiprime Rings


Published:20120127
Printed: Sep 2013
PaoKuei Liau,
Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan
ChengKai Liu,
Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan
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Abstract
Let $R$ be a semiprime ring with center
$Z(R)$. For $x,y\in R$, we denote by $[x,y]=xyyx$ the commutator of
$x$ and $y$. If $\sigma$ is a nonidentity automorphism of $R$ such
that
$$
\Big[\big[\dots\big[[\sigma(x^{n_0}),x^{n_1}],x^{n_2}\big],\dots\big],x^{n_k}\Big]=0
$$
for all $x \in R$, where $n_{0},n_{1},n_{2},\dots,n_{k}$ are fixed
positive integers, then there exists a map $\mu\colon R\rightarrow Z(R)$
such that $\sigma(x)=x+\mu(x)$ for all $x\in R$. In particular, when
$R$ is a prime ring, $R$ is commutative.