Given a local noetherian ring whose formal completion is integral, we introduce and study the -radical closure . Roughly speaking, this is the largest purely inseparable -subalgebra inside the formal completion . It turns out that the finitely generated intermediate rings 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.
"The $p$-radical closure of local noetherian rings." J. Commut. Algebra 12 (1) 135 - 151, Spring 2020. https://doi.org/10.1216/jca.2020.12.135