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, $R[X;\Gamma ]$ be the semigroup ring of Γ over R, and $R^t[X;\Gamma ]$ be the twisted semigroup ring of Γ over R with respect to t. In this paper, we show that $R^t[X;\Gamma ]$ is a GCD-domain if and only if R is a GCD-domain and Γ is a GCD-semigroup. Hence, $R^t[X;\Gamma ]$ is a GCD-domain if and only if $R[X;\Gamma ]$ is a GCD-domain, while $R[X;\Gamma ]$ need not be a UFD even though $R^t[X;\Gamma ]$ is a UFD. We show that if $\Gamma \cap -\Gamma =\left\{0\right\},$ then $R^t[X;\Gamma ]$ 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 $R^t[X;G]$ is a UFD.

Keywords:

• Twist function
• twisted semigroup ring
• GCD-domain
• ACCP
• UFD

2010 Mathematics Subject Classification:

• 13A02
• 13A15
• 13F15

Acknowledgments

We would like to thank the referee for his/her several valuable comments and suggestions.

Additional information

Funding

G. W. Chang was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B06029867), and D. Y. Oh was supported by Research Fund from Chosun University in 2017 and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07041083).