Topics in Commutative Ring Theory |
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Page 60
... module . Then there exists Χε Μ such that A / XA is annihilated by some power of M. The element X can be picked so as to avoid a given finite number of prime ideals M. To make this section self - contained we ... Macaulay modules 1 4 15 19.
... module . Then there exists Χε Μ such that A / XA is annihilated by some power of M. The element X can be picked so as to avoid a given finite number of prime ideals M. To make this section self - contained we ... Macaulay modules 1 4 15 19.
Page 63
... Macaulay modules The omission of Macaulay modules in CR should be remedied . The first step is to extend Theorem 138 to modules . R , Theorem 283. Let R be Noetherian , let P be prime in and let A be a faithful finitely generated R - module ...
... Macaulay modules The omission of Macaulay modules in CR should be remedied . The first step is to extend Theorem 138 to modules . R , Theorem 283. Let R be Noetherian , let P be prime in and let A be a faithful finitely generated R - module ...
Page 64
... module of grade 2. Thus R admits a faithful Macaulay module but is not Macaulay . ( 2 ) Following Nagata we can exhibit a three - dimensional We have local domain R of grade two which is not catenary . G ( A ) ≤ G ( R ) for all modules ...
... module of grade 2. Thus R admits a faithful Macaulay module but is not Macaulay . ( 2 ) Following Nagata we can exhibit a three - dimensional We have local domain R of grade two which is not catenary . G ( A ) ≤ G ( R ) for all modules ...
Common terms and phrases
1817 LIBRARIES A/XA a₁ analytically unramified annihilator antichains argument Artin-Rees lemma Artinian modules ascending chain condition assume closed set coefficients commutative ring contradiction countable deduce degree domain with quotient equation exists finite number finitely generated ideal finitely generated R-module graded ring height Hence homogeneous element homomorphic image ideals containing indeterminates induction integral closure integrally closed domain intersection isomorphic J-ideals Krull dimension M-primary Macaulay module maximal ideal MICHIG MICHIGAN CHIGAN minimal prime ideal neocommutative Noetherian ring non-zero-divisor number of elements number of prime P₁ partially ordered set Pick principal ideal principal prime proof of Theorem prove quasi-local quotient field R-sequence R₁ R₂ radical ideal regular local ring Remark sequence set of prime submodule subring subset Suppose Theorem 247 Theorem 306 two-dimensional regular U₁ UNIV UNIVERSITY valuation domain VERSITY Write X₁ zero-divisors ε Ι