Commutative RingsHandwritten notes were added in black ink by the author after publishing, some are corrections and some are additions. |
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Page 97
... Macaulay ring is a Noetherian ring in which any maximal ideal M satisfies rank ( M ) = grade ( M ) . Macaulay rings are the precise class of rings for which the unmixedness theorem holds . Theorem 8.8 . Let R be a Macaulay ring ; then ...
... Macaulay ring is a Noetherian ring in which any maximal ideal M satisfies rank ( M ) = grade ( M ) . Macaulay rings are the precise class of rings for which the unmixedness theorem holds . Theorem 8.8 . Let R be a Macaulay ring ; then ...
Page 100
... Macaulay ring , then little rank = rank . Exercises . 1. If R is a Macaulay ring , so is multiplicative subset S. S Rg for any then so is R ( x ) 2. If R is Macaulay for every maximal ideal , R. M 3. If R is Macaulay and x z ( R ) then is ...
... Macaulay ring , then little rank = rank . Exercises . 1. If R is a Macaulay ring , so is multiplicative subset S. S Rg for any then so is R ( x ) 2. If R is Macaulay for every maximal ideal , R. M 3. If R is Macaulay and x z ( R ) then is ...
Page 101
... Macaulay ring↔ R [ x ] ring . is a Macaulay Pick M maximal G ( P ) = rank ( P ) P , we may thus P is Proof . ← is proved by Ex . 3 above . ⇒ . in R [ x ] , then then P = RM is prime in R , is prime in R , say . Dividing out by a ...
... Macaulay ring↔ R [ x ] ring . is a Macaulay Pick M maximal G ( P ) = rank ( P ) P , we may thus P is Proof . ← is proved by Ex . 3 above . ⇒ . in R [ x ] , then then P = RM is prime in R , is prime in R , say . Dividing out by a ...
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Common terms and phrases
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