## Multiplicative Ideal Theory, Part 1 |

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### Contents

SOME BASIC CONCEPTS | 1 |

INTEGRAL EXTENSIONS OF RINGS | 77 |

VALUATION THEORY | 155 |

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### Common terms and phrases

abelian group algebraic cancellation ideal clear closure completely integrally closed Consequently consider contains a regular Corollary cp(N defined denote domain with identity domain with quotient extension finite set finite subset finitely generated ideal follows fractional ideals Galois group genuine prime ideal Gilmer Gilmer and Heinzer group G Hence holds idempotent implies induction integral dependence integral domain integral ideal integrally closed domain invertible ideal isolated subgroup isomorphism Jaffard Krull Lemma maximal ideal minimal prime monic polynomial monomials Nagata Noetherian P-adic P-primary positive integer Priifer domain primary ideals principal ideal PROOF properly contained Proposition prove quasi-local quotient field R-submodule rank regular element regular multiplicative system result ring of quotients ring with identity set of maximal set of prime shows subgroup of G subring suppose tion total quotient ring unique valuation ring value group zero divisors zero element zero monomials