Skew polynomial rings over $\sigma$-quasi-Baer and $\sigma$-principally quasi-Baer rings

Abstract

Let R be a ring R and ¾ be an endomorphism of R. R is called ¾-rigid (resp. reduced) if a¾(a) = 0 (resp. a2 = 0) for any a 2 R implies a = 0. An ideal I of R is called a ¾-ideal if ¾(I) µ I. R is called ¾-quasi-Baer (resp. right (or left) ¾-p:q:-Baer) if the right annihilator of every ¾-ideal (resp. right (or left) principal ¾-ideal) of R is generated by an idempotent of R. In this paper, a skew polynomial ring A = R[x; ¾] of a ring R is investigated as follows: For a ¾-rigid ring R, (1) R is ¾-quasi-Baer if and only if A is quasi-Baer if and only if A is ¹¾-quasi-Baer for every extended endomorphism ¹¾ on A of ¾; (2) R is right ¾-p.q.-Baer if and only if R is ¾-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 ¹¾-p.q.-Baer if and only if A is right ¹¾-p.q.-Baer for every extended endomorphism ¹¾ on A of ¾.