03171nam 2200577Ia 450 991048299850332120200520144314.03-642-31152-010.1007/978-3-642-31152-9(CKB)3400000000085875(SSID)ssj0000746090(PQKBManifestationID)11930881(PQKBTitleCode)TC0000746090(PQKBWorkID)10860443(PQKB)10605900(DE-He213)978-3-642-31152-9(MiAaPQ)EBC3070544(PPN)16532936X(EXLCZ)99340000000008587520120602d2012 uy 0engurnn|008mamaatxtccrPrime divisors and noncommutative valuation theory /Hidetoshi Marubayashi, Fred Van Oystaeyen1st ed. 2012.Berlin ;Heidelberg Springerc20121 online resource (IX, 218 p.) Lecture notes in mathematics,1617-9692 ;2059Bibliographic Level Mode of Issuance: Monograph3-642-31151-2 Includes bibliographical references and index.1. General Theory of Primes -- 2. Maximal Orders and Primes -- 3. Extensions of Valuations to some Quantized Algebras.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.Lecture notes in mathematics (Springer-Verlag) ;2059.Noncommutative ringsValuation theoryNoncommutative rings.Valuation theory.512.4616W4016W7016S3816H1013J2016T05mscMarubayashi Hidetoshi1941-477685Oystaeyen F. Van1947-55393MiAaPQMiAaPQMiAaPQBOOK9910482998503321Prime divisors and noncommutative valuation theory3372335UNINA