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A/xA algebraic annihilator assume Auslander-Buchsbaum chains of prime co-dimension coefficients commutative ring complete the proof contains a non-zero contradiction Corollary Dedekind ring defined Definition dimension direct summand discrete valuation ring elements Exercise exists Ext A,B finite free resolution finite number finitely generated R-module follows free module G-ideal G-ring Given grade hence Hilbert ring homomorphism hypothesis ideal contains induction integral domain integrally closed intersection invertible ideal Krull Lemma little rank Macaulay ring maximal chain maximal ideal maximal R-sequence minimal prime ideal Noetherian ring non-zero divisor Note P₁ P₁U polynomials principal ideal theorem principal prime projective prove Prüfer ring quotient field R-free R-projective rank Q regular local n-dimensional regular local ring result ring with quotient short exact sequences submodule subring torsion modules write x₁ zero zero-divisors χε