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Prime ideals and the Nullstellensatz
The ascending chain condition
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A/xA algebraic Ann(A annihilator assume Auslander-Buchsbaum chains of prime clearly co-dimension coefficients commutative ring complete the proof contains a non-zero contradiction Corollary Dedekind ring defined Definition dimension direct summand discrete valuation ring dR(A elements Exercise exists finite free resolution finite number finitely generated R-module follows free module G-ideal Given grade hence Hilbert ring hypothesis ideal Q induction integral domain integrally closed intersection invertible ideal Krull Lemma little rank Macaulay ring maximal chain maximal ideal maximal R-sequence minimal prime ideal modules with FFR Noetherian ring non-zero prime ideal Note polynomials principal ideal theorem principal prime projective prove PrUfer ring quotient field R-projective R*-free rank Q regular local n-dimensional regular local ring result ring with quotient short exact sequences submodule subring take x e torsion modules unmixedness theorem write x i z(A zero zero-divisors