Ebook: Manis Valuations and Prüfer Extensions I: A New Chapter in Commutative Algebra

The present book is devoted to a study of relative Prüfer rings and Manis valuations, with an eye to application in real and p-adic geometry. If one wants to expand on the usual algebraic geometry over a non-algebraically closed base field, e.g. a real closed field or p-adically closed field, one typically meets lots of valuation domains. Usually they are not discrete and hence not noetherian. Thus, for a further develomemt of real algebraic and real analytic geometry in particular, and certainly also rigid analytic and p-adic geometry, new chapters of commutative algebra are needed, often of a non-noetherian nature. The present volume presents one such chapter.

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The present book is devoted to a study of relative Pr?fer rings and Manis valuations, with an eye to application in real and p-adic geometry. If one wants to expand on the usual algebraic geometry over a non-algebraically closed base field, e.g. a real closed field or p-adically closed field, one typically meets lots of valuation domains. Usually they are not discrete and hence not noetherian. Thus, for a further develomemt of real algebraic and real analytic geometry in particular, and certainly also rigid analytic and p-adic geometry, new chapters of commutative algebra are needed, often of a non-noetherian nature. The present volume presents one such chapter.

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The present book is devoted to a study of relative Pr?fer rings and Manis valuations, with an eye to application in real and p-adic geometry. If one wants to expand on the usual algebraic geometry over a non-algebraically closed base field, e.g. a real closed field or p-adically closed field, one typically meets lots of valuation domains. Usually they are not discrete and hence not noetherian. Thus, for a further develomemt of real algebraic and real analytic geometry in particular, and certainly also rigid analytic and p-adic geometry, new chapters of commutative algebra are needed, often of a non-noetherian nature. The present volume presents one such chapter.
Content:
Introduction....Pages 1-6
Summary....Pages 7-7
Chapter I: Basics on Manis valuations and Pr?fer extensions....Pages 9-81
Chapter II: Multiplicative ideal theory....Pages 83-176
Chapter III: PM-valuations and valuations of weaker type....Pages 177-250
Appendix....Pages 251-256
References....Pages 257-262
Subject and Symbol Index....Pages 263-267

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The present book is devoted to a study of relative Pr?fer rings and Manis valuations, with an eye to application in real and p-adic geometry. If one wants to expand on the usual algebraic geometry over a non-algebraically closed base field, e.g. a real closed field or p-adically closed field, one typically meets lots of valuation domains. Usually they are not discrete and hence not noetherian. Thus, for a further develomemt of real algebraic and real analytic geometry in particular, and certainly also rigid analytic and p-adic geometry, new chapters of commutative algebra are needed, often of a non-noetherian nature. The present volume presents one such chapter.
Content:
Introduction....Pages 1-6
Summary....Pages 7-7
Chapter I: Basics on Manis valuations and Pr?fer extensions....Pages 9-81
Chapter II: Multiplicative ideal theory....Pages 83-176
Chapter III: PM-valuations and valuations of weaker type....Pages 177-250
Appendix....Pages 251-256
References....Pages 257-262
Subject and Symbol Index....Pages 263-267
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