Skew power series extensions of principally quasi-Baer rings

A ring R is called right principally quasi-Baer (or simply right p.q.-Baer ) if the right annihilator of a principal right ideal of R is generated by an idempotent. Let R be a ring such that all left semicentral idempotents are central. Let α be an endomorphism of R which is not assumed to be surjective and R be α -compatible. It is shown that the skew power series ring R [[ x; α ]] is right p.q.-Baer if and only if the skew Laurent power series ring R [[ x, x ⁻¹ ; α ]] is right p.q.-Baer if and only if R is right p.q.-Baer and any countable family of idempotents in R has a generalized join in I ( R ). An example showing that the α -compatible condition on R is not superfluous, is provided.