Quasi-duo and weakly duo skew PBW extensions

In this paper we study the quasi-duo and weakly duo properties for skew PBW extensions. We show that if [Formula: see text] is a right quasi-duo skew PBW extension over a ring [Formula: see text], then [Formula: see text] is right quasi-duo, and that for a right quasi-duo graded skew PBW extension [Formula: see text] over [Formula: see text], the zero-degree homogeneous component of [Formula: see text] is left quasi-duo if and only if [Formula: see text] is left quasi-duo. Moreover, we give a characterization of right quasi-duo rings for a particular kind of bijective quasi-commutative skew PBW extensions. We also show some properties of weakly left or right duo skew PBW extensions.