Divisibility properties of twisted semigroup rings. Issue 3 (3rd March 2020)

• Record Type: Journal Article
• Authors: Chang, Gyu Whan; Oh, Dong Yeol
• Is Part Of: Communications in algebra. Volume 48:Issue 3(2020)
• Page Start: 1191
• Page End: 1200
• Publication Date: 2020-03-03
• Subjects: Twist function -- twisted semigroup ring -- GCD-domain -- ACCP -- UFD; 13A02 -- 13A15 -- 13F15; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2019.1677693
• Languages: English
• ISSNs: 0092-7872

Abstract

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