Topics in Commutative Ring Theory |
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Page 60
... the appendix . So at present , I just state without proof a theorem that is needed . Theorem 281. Let R be local with maximal ideal M , A an Artinian R - module . Then there exists Χε Μ such that A / XA is annihilated by some power of M.
... the appendix . So at present , I just state without proof a theorem that is needed . Theorem 281. Let R be local with maximal ideal M , A an Artinian R - module . Then there exists Χε Μ such that A / XA is annihilated by some power of M.
Page 62
Irving Kaplansky. i - l X ( A / XA ) Hi ( A ) * H1 ( A ) . i Since A / XA has dimension k - 1 , the left term vanishes for Multiplication by X is thus one - to - one on i ≥ k + 1 . Hi ( A ) · X is locally nilpotent on the Artinian Hi ...
Irving Kaplansky. i - l X ( A / XA ) Hi ( A ) * H1 ( A ) . i Since A / XA has dimension k - 1 , the left term vanishes for Multiplication by X is thus one - to - one on i ≥ k + 1 . Hi ( A ) · X is locally nilpotent on the Artinian Hi ...
Page 71
Irving Kaplansky. Proof . axa a . Let Given a in R we must show that X S be the set of all elements ( a - ax1a ) ( a ... xa ) = = 0 , an ( 1 - za ) a " ( 1 - za ) = 0 , from which we deduce and permute by Theorem 293. The product ( 1 ...
Irving Kaplansky. Proof . axa a . Let Given a in R we must show that X S be the set of all elements ( a - ax1a ) ( a ... xa ) = = 0 , an ( 1 - za ) a " ( 1 - za ) = 0 , from which we deduce and permute by Theorem 293. The product ( 1 ...
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