Commutative Rings |
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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 dp(A elements Exercise exists Ext A,B finite free resolution finite number finitely generated R-module follows free module G-ideal Given grade hence Hilbert ring hypothesis ideal contains induction integral domain integrally closed intersection invertible ideal Krull Lemma little rank Macaulay ring maximal chain maximal ideal maximal primes belonging maximal R-sequence minimal prime ideal modules with FFR Noetherian ring non-zero prime ideal Note P₁ P₁U polynomials principal ideal theorem principal prime projective prove Prüfer ring quotient field R-free R-projective regular local n-dimensional regular local ring result ring with quotient short exact sequences submodule subring torsion modules write zero zero-divisors χε