## Commutative rings |

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### Contents

Noetherian Rings | 47 |

Macaulay Rings and Regular Rings | 84 |

Homological Aspects of Ring Theory | 123 |

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A/xA algebraic annihilator ascending chain condition assume coefficients commutative ring contradiction Dedekind deduce disjoint domain with quotient equation exact sequence FFR module finite number finitely generated non-zero finitely generated R-module G-domain G-ideal Gorenstein Hence Hilbert ring Hint hypothesis id(A idR(A implies induction integral domain integrally closed intersection invertible ideal isomorphic Jacobson radical Krull Let R C T Macaulay ring maximal ideal maximal sequence maximal with respect minimal prime ideal multiplicatively closed set Nakayama lemma Noetherian domain Noetherian ring non-zero element non-zero ideal non-zero prime ideal notation number of prime one-dimensional polynomial ring prime ideal containing principal ideal theorem principal prime proof of Theorem properly contained Prove quotient field R-sequence radical ideals rank(P regular local ring Remark ring with maximal satisfies the ascending submodule Suppose Theorem 142 Theorem 81 valuation domain write zero-divisor