Journal of the
Korean Mathematical Society JKMS

In this paper, we mainly discuss Gorenstein Dedekind domains (G-Dedekind domains for short) and their overrings. Let $R$ be a one-dimensional Noetherian domain with quotient field $K$ and integral closure $T$. Then it is proved that $R$ is a G-Dedekind domain if and only if for any prime ideal $P$ of $R$ which contains $(R:_KT)$, $P$ is Gorenstein projective. We also give not only an example to show that G-Dedekind domains are not necessarily Noetherian Warfield domains, but also a definition for a special kind of domain: a $2$-DVR. As an application, we prove that a Noetherian domain $R$ is a Warfield domain if and only if for any maximal ideal $M$ of $R$, $R_M$ is a $2$-DVR.

Keywords: Gorenstein projective module, Gorenstein Dedekind domain, strongly Gorenstein projective module, $n$-strongly Gorenstein projective module, Noetherian Warfield domain, 2-DVR

MSC numbers: 13G05, 13D03