The $p$-radical closure of local noetherian rings

Abstract

Given a local noetherian ring $R$ whose formal completion is integral, we introduce and study the $p$-radical closure $R^{prc}$. Roughly speaking, this is the largest purely inseparable $R$-subalgebra inside the formal completion $\widehat{R}$. It turns out that the finitely generated intermediate rings $R\subset A\subset R^{prc}$ have rather peculiar properties. They can be used in a systematic way to provide examples of integral local rings whose normalization is nonfinite, that do not admit a resolution of singularities, and whose formal completion is nonreduced.