Commutative RingsHandwritten notes were added in black ink by the author after publishing, some are corrections and some are additions. |
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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 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 hypothesis ideal contains induction integral domain integrally closed 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 divisor non-zero prime ideal Note P₁ polynomials principal ideal theorem principal prime projective prove Prüfer ring quotient field R-projective R*-free regular local n-dimensional regular local ring result ring with quotient short exact sequences submodule torsion modules write x₁ xz(A zero zero-divisors χε