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Prime Divisors and Noncommutative Valuation Theory

  • Book
  • © 2012

Overview

Authors:
  1. Hidetoshi Marubayashi

    1. Faculty of Science and Engineering, Tokushima Bunri University, Sanuki City, Kagawa, Japan

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  2. Fred Van Oystaeyen

    1. Mathematics and Computer Science, University of Antwerp, Antwerp, Belgium

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    You can also search for this author in PubMed   Google Scholar

  • Describes different generalizations of valuations from the common generalized concept of primes in algebras
  • Combines the arithmetic of finite dimensional central simple algebras with new theory for infinite dimensional ones
  • Mixing methods concerning value functions, valuation filtrations and orders over valuation rings for the study of new classes of algebras, e.g. quantized algebras, Weyl algebras and Hopf algebras
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2059)

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Table of contents (3 chapters)

  1. Front Matter

    Pages i-ix
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  2. General Theory of Primes

    • Hidetoshi Marubayashi, Fred Van Oystaeyen
    Pages 1-107

  3. Maximal Orders and Primes

    • Hidetoshi Marubayashi, Fred Van Oystaeyen
    Pages 109-173

  4. Extensions of Valuations to Quantized Algebras

    • Hidetoshi Marubayashi, Fred Van Oystaeyen
    Pages 175-211

  5. Back Matter

    Pages 213-218
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Keywords

About this book

Classical valuation theory has applications in number theory and class field theory as well as in algebraic geometry, e.g. in a divisor theory for curves.  But the noncommutative equivalent is mainly applied to finite dimensional skewfields.  Recently however, new types of algebras have become popular in modern algebra; Weyl algebras, deformed and quantized algebras, quantum groups and Hopf algebras, etc. The advantage of valuation theory in the commutative case is that it allows effective calculations, bringing the arithmetical properties of the ground field into the picture.  This arithmetical nature is also present in the theory of maximal orders in central simple algebras.  Firstly, we aim at uniting maximal orders, valuation rings, Dubrovin valuations, etc. in a common theory, the theory of primes of algebras.  Secondly, we establish possible applications of the noncommutative arithmetics to interesting classes of algebras, including the extension of central valuations to nice classes of quantized algebras, the development of a theory of Hopf valuations on Hopf algebras and quantum groups, noncommutative valuations on the Weyl field and interesting rings of invariants and valuations of Gauss extensions.

Authors and Affiliations

  • Faculty of Science and Engineering, Tokushima Bunri University, Sanuki City, Kagawa, Japan

    Hidetoshi Marubayashi

  • Mathematics and Computer Science, University of Antwerp, Antwerp, Belgium

    Fred Van Oystaeyen

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