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Rsequences and graded rings
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A/xA analytically unramified annihilator antichains argument Artin-Rees lemma Artinian modules ascending chain condition assume assumption chain of prime closed set coefficients commutative ring contradiction corollary countable deduce degree domain with quotient equation exists finite number finitely generated ideal finitely generated R-module formal multiple graded ring height(P Hence homogeneous element homomorphic image hypothesis ideals containing incomparable indeterminates induction integral closure integrally closed domain J-ideals Krull dimension lies M-primary Macaulay module maximal ideal minimal prime ideal neocommutative Noetherian ring non-zero-divisor number of elements number of prime partially ordered set Pick primary decomposition principal prime proof of Theorem prove quasi-local quotient field R C T R-sequence radical ideal regular local ring Remark residue class field set of prime submodule subring subset Suppose Theorem 293 Theorem 306 two-dimensional regular valuation domain whence Write zero-divisors