Let R be a ring R and ${sigma}$ be an endomorphism of R. R is called ${sigma}$-rigid (resp. reduced) if $a{sigma}r(a) = 0 (resp{cdot}a^2 = 0)$ 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 ${sigma}$.