Let R be a ring R and

be an endomorphism of R. R is called

-rigid (resp. reduced) if

for any

implies a = 0. An ideal I of R is called a

-ideal if

. R is called

-quasi-Baer (resp. right (or left)

-p.q.-Baer) if the right annihilator of every

-ideal (resp. right (or left) principal

-ideal) of R is generated by an idempotent of R. In this paper, a skew polynomial ring A = R[

] of a ring R is investigated as follows: For a

-rigid ring R, (1) R is

-quasi-Baer if and only if A is quasi-Baer if and only if A is

-quasi-Baer for every extended endomorphism

on A of

(2) R is right

-p.q.-Baer if and only if R is

-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

-p.q.-Baer if and only if A is right

-p.q.-Baer for every extended endomorphism

on A of

.