The Prime Spectrum of Commutative Differential Algebras

For a commutative differential algebra A, if p1 and p2 are prime ideals with p1 ⊂ p2, p1 ≠ p2, and D(p1) not contained in p2, then there exists a prime ideal p3 ⊂ p2, with p3 ⊄ p1 and p1 ⊄ p3 and {x/Dkx ∈ p1 for all k ≥ 0} ⊂ p3. This yields an alternative proof that there is no non-trivial derivation on Ck([0,1]) and gives a generalized existence theorem for polarized prime ideals of C∞([0,1], C).