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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.

Ordered algebraic structures --- Noncommutative rings. --- Valuation theory. --- Associative rings. --- Rings (Algebra). --- Algebra. --- Ordered algebraic structures. --- Category theory (Mathematics). --- Homological algebra. --- Commutative algebra. --- Commutative rings. --- Field theory (Physics). --- Associative Rings and Algebras. --- Order, Lattices, Ordered Algebraic Structures. --- Category Theory, Homological Algebra. --- Commutative Rings and Algebras. --- Field Theory and Polynomials. --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Rings (Algebra) --- Algebra --- Homological algebra --- Algebra, Abstract --- Homology theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Algebraic structures, Ordered --- Structures, Ordered algebraic --- Mathematics --- Mathematical analysis --- Algebraic rings --- Ring theory --- Algebraic fields --- Non-commutative rings --- Associative rings --- Algebraic number theory --- Topological fields

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Finite reductive groups and their representations lie at the heart of goup theory. After representations of finite general linear groups were determined by Green (1955), the subject was revolutionized by the introduction of constructions from l-adic cohomology by Deligne-Lusztig (1976) and by the approach of character-sheaves by Lusztig (1985). The theory now also incorporates the methods of Brauer for the linear representations of finite groups in arbitrary characteristic and the methods of representations of algebras. It has become one of the most active fields of contemporary mathematics. The present volume reflects the richness of the work of experts gathered at an international conference held in Luminy. Linear representations of finite reductive groups (Aubert, Curtis-Shoji, Lehrer, Shoji) and their modular aspects Cabanes Enguehard, Geck-Hiss) go side by side with many related structures: Hecke algebras associated with Coxeter groups (Ariki, Geck-Rouquier, Pfeiffer), complex reflection groups (Broué-Michel, Malle), quantum groups and Hall algebras (Green), arithmetic groups (Vignéras), Lie groups (Cohen-Tiep), symmetric groups (Bessenrodt-Olsson), and general finite groups (Puig). With the illuminating introduction by Paul Fong, the present volume forms the best invitation to the field.

Group theory --- Representations of groups --- Finite groups --- Congresses --- Group theory. --- Associative rings. --- Rings (Algebra). --- Algebraic geometry. --- Group Theory and Generalizations. --- Associative Rings and Algebras. --- Algebraic Geometry. --- Algebraic geometry --- Geometry --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Representations of groups - Congresses --- Finite groups - Congresses

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512.7 --- Algebraic geometry. Commutative rings and algebras --- 512.7 Algebraic geometry. Commutative rings and algebras --- Geometry, Algebraic --- Algebraic geometry --- Geometry

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Geometry, Algebraic. --- Hyperfunctions. --- Pseudodifferential operators. --- 517.95 --- 512.7 --- Partial differential equations --- Algebraic geometry. Commutative rings and algebras --- 512.7 Algebraic geometry. Commutative rings and algebras --- 517.95 Partial differential equations

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Ordered algebraic structures --- Projective modules (Algebra) --- Intersection theory. --- Intersection theory (Mathematics) --- Mathematical Theory --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Intersecties [Theorie van ] --- Intersection theory --- Intersections [Théorie des ] --- Modules [Projective ] (Algebra) --- Theorie van intersecties --- Théorie des intersections --- Commutative algebra. --- Commutative rings. --- Commutative Rings and Algebras. --- Rings (Algebra)