Weak dimension and right distributivity of skew generalized power series rings

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 power series rings, skew Laurent polynomial rings, skew group rings, and Mal'cev-Neumann Laurent series rings. In the case where S is positively ordered we give sufficient and necessary conditions for the skew generalized power series ring R[[S, ω]] to have weak dimension less than or equal to one. In particular, for such an S we show that the ring R[[S, ω]] is right duo of weak dimension at most one precisely when the lattice of right ideals of the ring R[[S, ω]] is distributive and ω(s) is injective for every s ∈ S.