Generators of D-modules in positive characteristic

Let R = k[x₁,..., x_d] or R = k[[x₁,..., x_d]] be either a polynomial or a formal power series ring in a finite number of variables over a field k of characteristic p > 0 and let D _R|k be the ring of k-linear differential operators of R. In this paper we prove that if f is a non-zero element of R then R_f, obtained from R by inverting f, is generated as a D_R|k-module by 1/f. This is an amazing fact considering that the corresponding characteristic zero statement is very false. In fact we prove an analog of this result for a considerably wider class of rings R and a considerably wider class of D _R|k-modules.