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# On Automorphisms and Commutativity in Semiprime Rings

Published:2012-01-27
Printed: Sep 2013
• Pao-Kuei Liau,
Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan
• Cheng-Kai 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 $[x,y]=xy-yx$ the commutator of $x$ and $y$. If $\sigma$ is a non-identity 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.
 Keywords: automorphism, generalized polynomial identity (GPI)
 MSC Classifications: 16N60 - Prime and semiprime rings [See also 16D60, 16U10] 16W20 - Automorphisms and endomorphisms 16R50 - Other kinds of identities (generalized polynomial, rational, involution)

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