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). 

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Abstract: 
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. 
Citation: 
DOI:10.3770/j.issn:20952651.2015.01.005 
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