Rings with Symmetric Endomorphisms and Their Extensions

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.