Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic

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Let R be a differentiably simple Noetherian commutative ring of characteristic p>0 (then (R,) is local with n:= emdim (R)<∞). A short proof is given of the Theorem of Harper Harper61 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 ᵖⁱ≠ 0 then ᵖⁱ(xᵢ)=1 for some xᵢ∈. The derivation is given explicitly, it is unique up to the action of the group Aut(R) of ring automorphisms of R. Let (R) be the set of all such derivations. Then (R)≃ Aut(R)/ Aut(R/). The proof is based on existence and uniqueness of an iterative - descent (for each ∈ (R)), i.e. a sequence {y[ⁱ], 0≤ i<pⁿ} in R such that y[⁰]:=1, (y[ⁱ])=y[ⁱ⁻¹] and y[ⁱ]y[ʲ]=i+j iy[ⁱ⁺ʲ] for all 0≤ i,j<pⁿ. For each ∈ (R), ₖ'(R)=⊕ᵢ₌₀ⁿ⁻¹Rᵖⁱ and k':= ()≃ R/.