Divisibility properties of twisted semigroup rings

Abstract

Let R be an integral domain, Γ be a nonzero torsion-free commutative cancellative monoid, t be a twist function of Γ on R, be the semigroup ring of Γ over R, and be the twisted semigroup ring of Γ over R with respect to t. In this paper, we show that is a GCD-domain if and only if R is a GCD-domain and Γ is a GCD-semigroup. Hence, is a GCD-domain if and only if is a GCD-domain, while need not be a UFD even though is a UFD. We show that if then is a UFD if and only if R is a UFD and Γ is a UFS. We also show that if G is a torsion-free abelian group satisfying the ascending chain condition on its cyclic subgroups, then R is a UFD if and only if is a UFD.