Ore Extensions of Extended Symmetric and Reversible Rings

Abstract

Let $\sigma$ be an endomorphism and $\delta$ an $\sigma$-derivation of a ring $R$. In this paper, we show that if $R$ is $(\sigma,\delta)$-skew Armendariz and $a\sigma(b)=0$ implies $ab=0$ for $a,b\in R$. Then $R$ is symmetric (respectively, reversible) if and only if $R$ is $\sigma$-symmetric (respectively, $\sigma$-reversible) if and only if $R[x;\sigma,\delta]$ is symmetric (respectively, reversible). Moreover, we study on the relationship between the Baerness, quasi-Baerness and p.q.-Baerness of a ring $R$ and these of the Ore extension $R[x;\sigma,\delta]$. As a consequence we obtain a partial generalization of \cite{hong/2000}.