On the purely inseparable closure of rings

Abstract:

Let $K \subseteqq R$ be commutative rings with identity 1. Let $D = \{ {D_i}\}$ be a higher derivation of $R$. We shall prove in this paper that if $K$ is invariant with respect to $D$, the purely inseparable closure $\overline {{K_R}}$ of $K$ in $R$ is invariant with respect to $D$ and the formal power series ring $\overline {{K_R}} [[t]]$ is purely inseparably closed in $R[[t]]$.