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Prime Divisors and Noncommutative Valuation Theory (Paperback, 2012 ed.)
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Prime Divisors and Noncommutative Valuation Theory (Paperback, 2012 ed.)
Series: Lecture Notes in Mathematics, 2059
Expected to ship within 10 - 15 working days
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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.
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