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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 .

SKEW POLYNOMIAL RINGS OVER σ-QUASI-BAER AND σ-PRINCIPALLY QUASI-BAER RINGS