Manis Valuations and Prüfer Extensions II

  • Manfred Knebusch
  • Tobias Kaiser

Part of the Lecture Notes in Mathematics book series (LNM, volume 2103)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Manfred Knebusch, Tobias Kaiser
    Pages 1-57
  3. Manfred Knebusch, Tobias Kaiser
    Pages 59-121
  4. Manfred Knebusch, Tobias Kaiser
    Pages 123-178
  5. Back Matter
    Pages 179-192

About this book

Introduction

This volume is a sequel to “Manis Valuation and Prüfer Extensions I,” LNM1791. The Prüfer extensions of a commutative ring A are roughly those commutative ring extensions R / A,where commutative algebra is governed by Manis valuations on R with integral values on A. These valuations then turn out to belong to the particularly amenable subclass of PM (=Prüfer-Manis) valuations. While in Volume I Prüfer extensions in general and individual PM valuations were studied, now the focus is on families of PM valuations. One highlight is the presentation of a very general and deep approximation theorem for PM valuations, going back to Joachim Gräter’s work in 1980, a far-reaching extension of the classical weak approximation theorem in arithmetic. Another highlight is a theory of so called “Kronecker extensions,” where PM valuations are put to use in  arbitrary commutative  ring extensions in a way that ultimately goes back to the work of Leopold Kronecker.

Keywords

Approximation theorems Kronecker extensions Manis valuations and Prüfer extensions Multiplicative ideal theory Star operations

Authors and affiliations

  • Manfred Knebusch
    • 1
  • Tobias Kaiser
    • 2
  1. 1.Fakultät f.MathematikUniversität RegensburgRegensburgGermany
  2. 2.Fakultät f.Informatik u.MathematikUniversität PassauPassauGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-03212-2
  • Copyright Information Springer International Publishing Switzerland 2014
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-03211-5
  • Online ISBN 978-3-319-03212-2
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • About this book