Prime Divisors and Noncommutative Valuation Theory / by Hidetoshi Marubayashi, Fred Van Oystaeyen.

Format:

Book

Author/Creator:

Marubayashi, Hidetoshi, 1941- author.

Van Oystaeyen, Fred, author.

Contributor:

SpringerLink (Online service)

Series:

Lecture Notes in Mathematics, 0075-8434 ; 2059.

Lecture Notes in Mathematics, 0075-8434 ; 2059

Language:

English

Subjects (All):

Algebra.

Geometry.

Geometry, Algebraic.

Algebraic Geometry.

Associative Rings and Algebras.

Local Subjects:

Algebra.

Geometry.

Algebraic Geometry.

Associative Rings and Algebras.

Physical Description:

1 online resource (IX, 218 pages).

Contained In:

Springer eBooks

Place of Publication:

Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2012.

System Details:

text file PDF

Summary:

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, et cetera 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, et cetera 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.

Contents:

1. General Theory of Primes

2. Maximal Orders and Primes

3. Extensions of Valuations to some Quantized Algebras.

Other Format:

Printed edition:

ISBN:

9783642311529

Access Restriction:

Restricted for use by site license.