On the purely inseparable closure of rings

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