Duo, Bézout and distributive rings of skew power series

We give necessary and sufficient conditions on a ring R and an endomorphism σ of R for the skew power series ring R[[x; σ]] to be right duo right Bézout. In particular, we prove that R[[x; σ]] is right duo right Bézout if and only if R[[x; σ]] is reduced right distributive if and only if R[[x; σ]] is right duo of weak dimension less than or equal to 1 if and only if R is N0-injective strongly regular and σ is bijective and idempotent-stabilizing, extending to skew power series rings the Brewer-Rutter-Watkins characterization of commutative B´ezout power series rings.