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035   $a(DE-He213)978-3-642-31152-9
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100   $a20120821d2012      u|             0
101 0 $aeng
135   $aurnn|008mamaa
181   $ctxt
182   $cc
183   $acr
200 10$aPrime Divisors and Noncommutative Valuation Theory$b[electronic resource] /$fby Hidetoshi Marubayashi, Fred Van Oystaeyen
205   $a1st ed. 2012.
210  1$aBerlin, Heidelberg :$cSpringer Berlin Heidelberg :$cImprint: Springer,$d2012.
215   $a1 online resource (IX, 218 p.) 
225 1 $aLecture Notes in Mathematics,$x0075-8434 ;$v2059
300   $aBibliographic Level Mode of Issuance: Monograph
311   $a3-642-31151-2 
320   $aIncludes bibliographical references and index.
327   $a1. General Theory of Primes -- 2. Maximal Orders and Primes -- 3. Extensions of Valuations to some Quantized Algebras.
330   $aClassical valuation theory has applications in number theory and class field theory as well as in algebraic geometry, e.g. in a divisor theory for curves.  But the noncommutative equivalent is mainly applied to finite dimensional skewfields.  Recently however, new types of algebras have become popular in modern algebra; Weyl algebras, deformed and quantized algebras, quantum groups and Hopf algebras, etc. The advantage of valuation theory in the commutative case is that it allows effective calculations, bringing the arithmetical properties of the ground field into the picture.  This arithmetical nature is also present in the theory of maximal orders in central simple algebras.  Firstly, we aim at uniting maximal orders, valuation rings, Dubrovin valuations, etc. in a common theory, the theory of primes of algebras.  Secondly, we establish possible applications of the noncommutative arithmetics to interesting classes of algebras, including the extension of central valuations to nice classes of quantized algebras, the development of a theory of Hopf valuations on Hopf algebras and quantum groups, noncommutative valuations on the Weyl field and interesting rings of invariants and valuations of Gauss extensions.
410  0$aLecture Notes in Mathematics,$x0075-8434 ;$v2059
606   $aAlgebra
606   $aGeometry
606   $aAlgebraic geometry
606   $aAssociative rings
606   $aRings (Algebra)
606   $aAlgebra$3https://scigraph.springernature.com/ontologies/product-market-codes/M11000
606   $aGeometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M21006
606   $aAlgebraic Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M11019
606   $aAssociative Rings and Algebras$3https://scigraph.springernature.com/ontologies/product-market-codes/M11027
615  0$aAlgebra.
615  0$aGeometry.
615  0$aAlgebraic geometry.
615  0$aAssociative rings.
615  0$aRings (Algebra).
615 14$aAlgebra.
615 24$aGeometry.
615 24$aAlgebraic Geometry.
615 24$aAssociative Rings and Algebras.
676   $a512/.46
686   $a16W40$a16W70$a16S38$a16H10$a13J20$a16T05$2msc
700   $aMarubayashi$b Hidetoshi$4aut$4http://id.loc.gov/vocabulary/relators/aut$0477685
702   $aVan Oystaeyen$b Fred$4aut$4http://id.loc.gov/vocabulary/relators/aut
906   $aBOOK
912   $a996466476303316
996   $aPrime Divisors and Noncommutative Valuation Theory$92831069
997   $aUNISA