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A/xA algebraic annihilator apply assume belonging called clearly co-dimension coefficients commutative ring condition consider contains contradiction Conversely Corollary Dedekind Dedekind domain defined Definition dimension direct elements equality equations equivalent exact sequence Example Exercise exists fact finite free resolution finitely generated R-module follows G-ring Given gives grade hence holds homomorphism hypothesis induction integral domain integrally closed intersection invertible ideal Krull Lemma length Macaulay ring maximal ideal maximal R-sequence module multiplicative natural Noetherian ring non-zero divisor Note obtain otherwise particular Pick polynomials possible prime ideal principal ideal theorem projective Proof prove quotient field R-projective R-sequence rank regular local ring relation Remark Resolve result satisfies short splits submodule Suppose Theorem torsion modules unique unit valuation ring variety whence write zero