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© 2014

Manis Valuations and Prüfer Extensions II

Book

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

  1. 1.Fakultät f.MathematikUniversität RegensburgRegensburgGermany
  2. 2.Fakultät f.Informatik u.MathematikUniversität PassauPassauGermany

Bibliographic information

  • Book Title Manis Valuations and Prüfer Extensions II
  • Authors Manfred Knebusch
    Tobias Kaiser
  • Series Title Lecture Notes in Mathematics
  • Series Abbreviated Title Lect.Notes Mathematics
  • 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 Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-319-03211-5
  • eBook ISBN 978-3-319-03212-2
  • Series ISSN 0075-8434
  • Series E-ISSN 1617-9692
  • Edition Number 1
  • Number of Pages XII, 190
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Commutative Rings and Algebras
  • Buy this book on publisher's site