The ascending chain condition for principal left or right ideals of skew generalized power series rings

Let R be a ring, S a monoid and a monoid homomorphism. In this paper we prove that if the monoid S is strictly totally ordered or S is commutative torsion-free cancellative semisubtotally ordered, then the ring R〚S,ω〛 of skew generalized power series with coefficients in R and exponents in S is a domain satisfying the ascending chain condition on principal left (resp. right) ideals if and only if R is a domain, R and S satisfy the ascending chain condition on principal left (resp. right) ideals and each ω(s) is injective (resp. is injective and preserves nonunits of R). As an immediate consequence we obtain characterizations of power series rings, Laurent series rings, skew power series rings, skew Laurent series rings and generalized power series rings that are domains satisfying the ascending chain condition on principal left (or right) ideals. We construct examples of skew generalized power series domains for which the ascending chain conditions on principal one-sided ideals are not symmetric.