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＜E-Book＞
Non-Commutative Valuation Rings and Semi-Hereditary Orders / by H. Marubayashi, Haruo Miyamoto, Akira Ueda
(K-Monographs in Mathematics ; 3)

Edition	1st ed. 1997.
Publisher	(Dordrecht : Springer Netherlands : Imprint: Springer)
Year	1997
Language	English
Size	VIII, 192 p : online resource
Authors	*Marubayashi, H author Miyamoto, Haruo author Ueda, Akira author SpringerLink (Online service)
Subjects	LCSH:Associative rings LCSH:Associative algebras LCSH:Algebra LCSH:Algebra, Homological LCSH:Commutative algebra LCSH:Commutative rings LCSH:Algebraic fields LCSH:Polynomials FREE:Associative Rings and Algebras FREE:Order, Lattices, Ordered Algebraic Structures FREE:Category Theory, Homological Algebra FREE:Commutative Rings and Algebras FREE:Field Theory and Polynomials
Notes	I. Semi-Hereditary and Prüfer Orders -- II. Dubrovin Valuation Rings -- III. Semi-Local Bezout Orders -- IV. The Applications and Examples -- A1. Semi-perfect rings and serial rings -- A2. Coherent rings -- A3. Azumaya algebras -- A4. The lifting idempotents -- A5. Wedderburn’s Theorem -- References -- Index of Notation Much progress has been made during the last decade on the subjects of non­ commutative valuation rings, and of semi-hereditary and Priifer orders in a simple Artinian ring which are considered, in a sense, as global theories of non-commu­ tative valuation rings. So it is worth to present a survey of the subjects in a self-contained way, which is the purpose of this book. Historically non-commutative valuation rings of division rings were first treat­ ed systematically in Schilling's Book [Sc], which are nowadays called invariant valuation rings, though invariant valuation rings can be traced back to Hasse's work in [Has]. Since then, various attempts have been made to study the ideal theory of orders in finite dimensional algebras over fields and to describe the Brauer groups of fields by usage of "valuations", "places", "preplaces", "value functions" and "pseudoplaces". In 1984, N. 1. Dubrovin defined non-commutative valuation rings of simple Artinian rings with notion of places in the category of simple Artinian rings and obtained significant results on non-commutative valuation rings (named Dubrovin valuation rings after him) which signify that these rings may be the correct def­ inition of valuation rings of simple Artinian rings. Dubrovin valuation rings of central simple algebras over fields are, however, not necessarily to be integral over their centers HTTP:URL=https://doi.org/10.1007/978-94-017-2436-4

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