On the PQ-Baer Skew Generalized Power Series Modules

For a ring R and a strictly totally ordered monoid (S,<=), let omega: S -> End(R) be a monoid homomorphism and M an (S,omega)-weakly rigid right R-module (i.e., for any elements m is an element of M, b is an element of R and s is an element of S, mRb=0 if and only if m omega(s)(Rb)=0), where End(R) is the ring of ring endomorphisms of R. It is shown that the skew generalized power series module M[[S]](R[[S,omega]]) is a principally quasi-Baer module if and only if the annihilator of every submodule generated by an S-indexed subset of M is generated by an idempotent as a right ideal of R. As a consequence we deduce that for an (S,omega)-weakly rigid ring R, the skew generalized power series ring R[[S,omega]] is right principally quasi-Baer if and only if R is right principally quasi-Baer and any S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join in R. The range of previous results in this area is expanded by these results.