Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic

Let R be a differentiably simple Noetherian commutative ring of characteristic p > 0 (then (R,m) is local with n := emdim(R) < ∞). A short proof is given of the Theorem of Harper [1] on classification of differentiably simple Noetherian commutative rings in prime characteristic. The main result of the paper is that there exists a nilpotent simple derivation of the ring R such that if δ i 6= 0 then δ i (xi) = 1 for some xi ∈ m. The derivation δ is given explicitly, it is unique up to the action of the group Aut(R) of ring automorphisms of R. Let nsder(R) be the set of all such derivations. Then nsder(R) ≃ Aut(R)/Aut(R/m). The proof is based on existence and uniqueness of an iterative δ-descent (for each δ ∈ nsder(R)), i.e. a sequence {y[i], 0 ≤ i < p} in R such that y[0] := 1, δ(y[i]) = y[i−1] and y[i]y[j] = ( i+j i ) y[i+j] for all 0 ≤ i, j < p. For each δ ∈ nsder(R), Derk′(R) = ⊕ n−1 i=0 Rδ p and k′ := ker(δ) ≃ R/m.