Lecture Notes in Mathematics

Prime Divisors and Noncommutative Valuation Theory

Authors: Marubayashi, Hidetoshi, Van Oystaeyen, Fred

  • 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
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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.

Table of contents (3 chapters)

  • General Theory of Primes

    Marubayashi, Hidetoshi (et al.)

    Pages 1-107

  • Maximal Orders and Primes

    Marubayashi, Hidetoshi (et al.)

    Pages 109-173

  • Extensions of Valuations to Quantized Algebras

    Marubayashi, Hidetoshi (et al.)

    Pages 175-211

Buy this book

eBook $44.99
price for USA (gross)
  • ISBN 978-3-642-31152-9
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $59.95
price for USA
  • ISBN 978-3-642-31151-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Prime Divisors and Noncommutative Valuation Theory
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2059
Copyright
2012
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-31152-9
DOI
10.1007/978-3-642-31152-9
Softcover ISBN
978-3-642-31151-2
Series ISSN
0075-8434
Edition Number
1
Number of Pages
IX, 218
Topics