On the P.Q.-Baer Skew Generalized Power Series Modules

Abstract

For a ring [Formula: see text] and a strictly totally ordered monoid [Formula: see text], let [Formula: see text] be a monoid homomorphism and [Formula: see text] an [Formula: see text]-weakly rigid right [Formula: see text]-module (i.e., for any elements [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text]), where [Formula: see text] is the ring of ring endomorphisms of [Formula: see text]. It is shown that the skew generalized power series module [Formula: see text] is a principally quasi-Baer module if and only if the annihilator of every submodule generated by an [Formula: see text]-indexed subset of [Formula: see text] is generated by an idempotent as a right ideal of [Formula: see text]. As a consequence we deduce that for an [Formula: see text]-weakly rigid ring [Formula: see text], the skew generalized power series ring [Formula: see text] is right principally quasi-Baer if and only if [Formula: see text] is right principally quasi-Baer and any [Formula: see text]-indexed subset of right semicentral idempotents in [Formula: see text] has a generalized [Formula: see text]-indexed join in [Formula: see text]. The range of previous results in this area is expanded by these results.