Left principally quasi-Baer and left APP-rings of skew generalized power series

Abstract

Let R be a ring, S a strictly ordered monoid, and ω : S → End (R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, and (skew) monoid rings. We characterize when a skew generalized power series ring R[[S, ω]] is left principally quasi-Baer and under various finiteness conditions on R we characterize when the ring R[[S, ω]] is left APP. As immediate corollaries we obtain characterizations for all aforementioned classical ring constructions to be left principally quasi-Baer or left APP. Such a general approach not only gives new results for several constructions simultaneously, but also serves the unification of already known results.