Divisibility Properties of Group Rings over Torsion-free Abelian Groups

Let G be a torsion-free abelian group of type (0, 0, 0, . . . ) and R an integrally closed integral domain with quotient field K. We show that every divisorial ideal (respectively, t-ideal) J of the group ring R[X;G] is of the form J = hIR[X;G] for some h ∈ K[X;G] and a divisorial ideal (respectively, t-ideal) I of R. Consequently, there are natural monoid isomorphisms Cl(R) ∼= Cl(R[X;G]) and Clt(R) ∼= Clt(R[X;G]). Throughout this paper, G shall stand for a torsion-free abelian group, R an integral domain, and K quotient field of R. Gilmer and Parker [5, Theorem 7.13] proved that R[X;G] is a unique factorization domain (UFD) if and only if R is a UFD and G is of type (0, 0, 0, . . . ). Using this result, Matsuda[8, Proposition 3.3] proved that R[X;G] is a Krull domain if and only if R is a Krull domain and G is of type (0, 0, 0, . . . ). In [9, Proposition 1 and 6 §12], Matsuda proved that if R[X;G] is a π-domain, then R is a π-domain and G is of type (0, 0, 0, . . . ). Later, in [1, Proposition 6.5], D.D. Anderson and D.F. Anderson proved that R[X;G] is a π-domain if and only if R is a π-domain and G is of type (0, 0, 0, . . . ). In [2, Proposition 6.5], it was also shown that the semigroup ring R[X; Γ] is a PVMD if and only if R is a PVMD and Γ is a PVMD semigroup. This is an answer to Malik’s question [7]. In [7], Malik studied divisorial properties of the semigroup ring R[X; Γ]. One of her results is that if R is an integrally closed domain and J is a divisorial ideal of the semigroup ring R[X; Γ] such that J ∩R 6= (0), then J is extended