Dp-minimal integral domains

Abstract

It is shown that every dp-minimal integral domain $R$ is a local ring and for every non-maximal prime ideal $\mathfrak{p}$ of $R$, the localization $R_{\mathfrak{p}}$ is a valuation ring and $\mathfrak{p}{R}_{\mathfrak{p}}=\mathfrak{p}$. Furthermore, a dp-minimal integral domain is a valuation ring if and only if its residue field is infinite or its residue field is finite and its maximal ideal is principal.

16 pages