 Rings with Symmetric Endomorphisms and Their Extensions
Received:May 13, 2014  Revised:June 23, 2014
Key Word: symmetric $\alpha$-ring   weak symmetric $\alpha$-ring   polynomial extension   classical quotient ring extension   Ore extension
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11101217) and the Natural Science Foundation of Jiangsu Province (Grant No.BK20141476).
 Author Name Affiliation Yao WANG School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Jiangsu 210044, P. R. China Weiliang WANG School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, P. R. China Yanli REN School of Mathematics and Information Technology, Nanjing Xiaozhuang University, Jiangsu 211171, P. R. China
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Let $R$ be a ring with an endomorphism $\alpha$ and an $\alpha$-derivation $\delta$. We introduce the notions of symmetric $\alpha$-rings and weak symmetric $\alpha$-rings which are generalizations of symmetric rings and weak symmetric rings, respectively, discuss the relations between symmetric $\alpha$-rings and related rings and investigate their extensions. We prove that if $R$ is a reduced ring and $\alpha(1)=1$, then $R$ is a symmetric $\alpha$-ring if and only if $R[x]/(x^{n})$ is a symmetric $\bar{\alpha}$-ring for any positive integer $n$. Moreover, it is proven that if $R$ is a right Ore ring, $\alpha$ an automorphism of $R$ and $Q(R)$ the classical right quotient ring of $R$, then $R$ is a symmetric $\alpha$-ring if and only if $Q(R)$ is a symmetric $\bar{\alpha}$-ring. Among others we also show that if a ring $R$ is weakly $2$-primal and $(\alpha,\delta)$-compatible, then $R$ is a weak symmetric $\alpha$-ring if and only if the Ore extension $R[x;\alpha,\delta]$ of $R$ is a weak symmetric $\bar{\alpha}$-ring.