Duo, Bézout, and Distributive Rings of Skew Power Series

Abstract

We give necessary and sufficient conditions on a ring $R$ and an endomorphism $\sigma$ of $R$ for the skew power series ring $R[[x; \sigma]]$ to be right duo right Bézout. In particular, we prove that $R[[x; \sigma]]$ is right duo right Bézout if and only if $R[[x; \sigma]]$ is reduced right distributive if and only if $R[[x; \sigma]]$ is right duo of weak dimension less than or equal to $1$ if and only if $R$ is $\aleph_0$-injective strongly regular and $\sigma$ is bijective and idempotent-stabilizing, extending to skew power series rings the Brewer-Rutter-Watkins characterization of commutative Bézout power series rings.