The $p$-radical closure of local noetherian rings

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 R ̂ . It turns out that the finitely generated intermediate rings R ⊂ A ⊂ 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.