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Prime Divisors and Noncommutative Valuation Theory

English · Paperback / Softback

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Description

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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.

List of contents

1. General Theory of Primes.- 2. Maximal Orders and Primes.- 3. Extensions of Valuations to some Quantized Algebras

Product details

Authors Hidetoshi Marubayashi, Fred Van Oystaeyen, Fred Van Oystaeyen
Publisher Springer, Berlin
 
Languages English
Product format Paperback / Softback
Released 22.08.2012
 
EAN 9783642311512
ISBN 978-3-642-31151-2
No. of pages 218
Dimensions 156 mm x 235 mm x 234 mm
Weight 362 g
Illustrations IX, 218 p.
Series Lecture Notes in Mathematics
Subjects Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra

Algebra, Geometrie, B, Algebraische Geometrie, geometry, Mathematics and Statistics, Algebraic Geometry, Rings (Algebra), Associative rings, Associative Rings and Algebras

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