$\boldsymbol{\mathit{m}}$-adic $p$-basis and regular local ring

We introduce the concept of $\boldsymbol{\mathit{m}}$-adic $p$-basis as an extension of the concept of $p$-basis. Let $(S,\boldsymbol{\mathit{m}})$ be a regular local ring of prime characteristic $p$ and $R$ a ring such that $S \supset R \supset S^p$. Then we prove that $R$ is a regular local ring if and only if there exists an $\boldsymbol{\mathit{m}}$-adic $p$-basis of $S/R$ and $R$ is Noetherian.