Principal Quasi-Baerness of Modules of Generalized Power Series

Abstract

Let $R$ be a ring, $M$ a right $R$-module and $(S,\leq)$ a strictly totally ordered monoid. It is shown that $[[M^{S,\leq}]]$, the module of generalized power series with coefficients in $M$ and exponents in $S$, is a p.q.Baer right $[[R^{S,\leq}]]$-module if and only if the right annihilator of any $S$-indexed family of cyclic submodules of $M$ in $R$ is generated by an idempotent of $R$. Furthermore, we will show that for a ring $R$ with all left semicentral idempotents are central, the ring $[[R^{S,\leq}]]$ consisting of generalized power series over $R$ is a right p.q.Baer ring if and only if $R$ is a right p.q.Baer ring and any $S$-indexed family of central idempotents of $R$ has a generalized join in $I(R)$, where $I(R)$ is the set of all idempotents of $R$.