n-adic p-basis and Regular semi-local ring

Abstract

Let $(R,\text{m})$ be a regular local ring of prime characteristic $p$ and $R\text{'}$ a Noetherian subring of $R$ such that $R^p\subset R\text{'}$. In the previous paper Furuya and Niitsuma [1], introducing the concept of $\text{m}$-adic $p$-basis, the first and the second authors proved the following theorem: The regular local ring $R$ has an $\text{m}$-adic $p$-basis over the subring $R\text{'}$ if and only if $R\text{'}$ is a regular local ring. In this paper, we generalize this result to semi-local rings.