(t, v)-Dedekind Domains and the Ring $${{R[X]_{N_v}}}$$

Abstract

Let R be an integral domain and X an indeterminate over R. In this paper, we indicate that the quotient ring of a (t, v)-Dedekind domain is not necessarily a (t, v)-Dedekind domain. Also, we show that a locally (t, v)-Dedekind domain is not necessarily a (t, v)-Dedekind domain. The characterization of the localization of a (t, v)-Dedekind domain further leads us to study the quotient ring \({R[X]_{N_v}}\) over a (t, v)-Dedekind domain R. As the application of the ring \({R[X]_{N_v}}\), we end this paper by characterizing the group ring R[X; G] and the semigroup ring R[Γ] over a (t, v)-Dedekind domain R.