Journal of the
Korean Mathematical Society JKMS

Let $R$ be a ring $R$ and $\sigma$ be an endomorphism of $R$. $R$ is called $\sigma$-rigid (resp. reduced) if $a\sigma(a) = 0$ (resp. $a^2 = 0$) for any $a\in R$ implies $a = 0$. An ideal $I$ of $R$ is called a $\sigma$-ideal if $\sigma(I) \subseteq$ $I$. $R$ is called $\sigma$-quasi-Baer (resp. right (or left) $\sigma$-$p$. $q$.-Baer) if the right annihilator of every $\sigma$-ideal (resp. right (or left) principal $\sigma$-ideal) of $R$ is generated by an idempotent of $R$. In this paper, a skew polynomial ring $A = R[x; \sigma]$ of a ring $R$ is investigated as follows: For a $\sigma$-rigid ring $R$, (1) $R$ is $\sigma$-quasi-Baer if and only if $A$ is quasi-Baer if and only if $A$ is $\sigma$-quasi-Baer for every extended endomorphism $\sigma$ on $A$ of $\sigma$; (2) $R$ is right $\sigma$-p.q.-Baer if and only if $R$ is $\sigma$-p.q.-Baer if and only if $A$ is right p.q.-Baer if and only if $A$ is p.q.-Baer if and only if $A$ is $\sigma$-p.q.-Baer if and only if $A$ is right $\sigma$-p.q.-Baer for every extended endomorphism $\sigma$ on $A$ of $\sigma$.

Keywords: $\sigma$-rigid ring, $\sigma$-Baer, $\sigma$-quasi-Baer, $\sigma$-p.q.-Baer ring, $\sigma$-p.p. ring, skew polynomial ring

MSC numbers: 16S36