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1-dimensional Cohen-Macaulay ring Artinian divisible R-module Artinian R-module assume canonical ideal closed component commutative ring completion composition series Corollary 2.5 cotorsion R-module define direct sum discrete valuation ring domain of Krull epimorphism exact sequence ExtR(K,B finite length finitely generated R-module full ring Gorenstein ring H 8R Q h-divisible h-reduced hdRQ Hence by Theorem homomorphic image idea1 ideal of H ideal of rank injective R-module integral closure integral domain isomorphic Ker f Krull dimension Lemma M-primary ideal maximal ideal Noetherian local domain Noetherian ring one-to-one correspondence P-primary P-primary component prime ideal pseudo valuation rings Q 8R quasi-local ring R-homomorphism R-topology regular element regular ideal ring of quotients series of divisible simple divisible R-module simple divisible submodule statements are equivalent strongly unramified ring superficial element superficial of degree Suppose Theorem 6.6 torsion R-module torsion-free and divisible torsion-free R-module unmixed ideal unramified ring extension zero divisors