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Manis valuations and Prüfer extensions I : a new chapter in commutative algebra /

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The present book is devoted to a study of relative Prfer 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 typ...

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Bibliographic Details
Main Author: Knebusch, Manfred
Corporate Author: SpringerLink (Online service)
Other Authors: Zhang, Digen, 1965-
Format: eBook
Language:English
Published: Berlin ; New York : Springer, [2002]
Series:Lecture notes in mathematics (Springer-Verlag) ; 1791.
Subjects:
Prüfer rings.
Graph labelings.
Commutative algebra.
Electronic books.
Online Access:Connect to the full text of this electronic book
Table of Contents:
  • Ch. I. Basics on Manis valuations and Prufer extensions
  • 1. Valuations of rings
  • 2. Valuation subrings and Manis pairs
  • 3. Weakly surjective homomorphisms
  • 4. More on weakly surjective extensions
  • 5. Basic theory of Prufer extensions
  • 6. Examples of Prufer extensions and convenient ring extensions
  • 7. Principal ideal results
  • Ch. II. Multiplicative ideal theory
  • 1. Multiplicative properties of regular modules
  • 2. Characterizing Prufer extensions' by the behavior of their regular ideals
  • 3. Describing a Prufer extension by its lattice of regular ideals
  • 4. Tight extensions
  • 5. Distributive submodules
  • 6. Transfer theorems
  • 7. Polars and factors in a Prufer extension
  • 8. Decomposition of regular modules
  • 9. Prufer overmodules
  • 10. Bezout extensions
  • 11. The Prufer extensions of a noetherian ring
  • 12. Invertible hulls for modules over noetherian rings
  • Ch. III. PM-valuations and valuations of weaker type
  • 1. The PM-overrings in a Prufer extension
  • 2. Regular modules in a PM-extension
  • 3. More ways to characterize PM-extensions, and a look at BM-extensions
  • 4. Tight valuations
  • 5. Existence of various valuation hulls
  • 6. Inside and outside the Manis valuation hull
  • 7. The TV-hull in a valuative extension
  • 8. Principal valuations
  • 9. Descriptions of the PM-hull
  • 10. Composing valuations with ring homomorphisms
  • 11. Transfer of valuations
  • App. A. (to I, Session 4 and I, Session 5): Flat epimorphisms
  • App. B. (to II, Session 2): Arithmetical rings
  • App. C. (to III, Session 6): A direct proof of the existence of Manis valuation hulls.