Ebook: Non-Commutative Valuation Rings and Semi-Hereditary Orders

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.

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This is the first self-contained summary of non-Noetherian orders in a simple Artinian ring, a subject in which much progress has been made in the last decade.
The contents of the book are mainly Dubrovin valuation rings and semi-hereditary orders, including Pr?fer and semi-local Bezout orders, which are considered, in a sense, as global theories of Dubrovin valuation rings. These are then developed further, and applied to give some examples such as Dubrovin valuation rings in crossed product algebras, semi-hereditary maximal order in certain matrix rings, and the idealizers of semi-hereditary orders and Henselization of Bezout orders.
Audience: This volume will be of interest to researchers and graduate students whose work involves non-commutative ring theory and module theory.

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This is the first self-contained summary of non-Noetherian orders in a simple Artinian ring, a subject in which much progress has been made in the last decade.
The contents of the book are mainly Dubrovin valuation rings and semi-hereditary orders, including Pr?fer and semi-local Bezout orders, which are considered, in a sense, as global theories of Dubrovin valuation rings. These are then developed further, and applied to give some examples such as Dubrovin valuation rings in crossed product algebras, semi-hereditary maximal order in certain matrix rings, and the idealizers of semi-hereditary orders and Henselization of Bezout orders.
Audience: This volume will be of interest to researchers and graduate students whose work involves non-commutative ring theory and module theory.
Content:
Front Matter....Pages i-viii
Semi-Hereditary and Pr?fer Orders....Pages 1-20
Dubrovin Valuation Rings....Pages 21-78
Semi-Local Bezout Orders....Pages 79-108
The Applications and Examples....Pages 109-173
Back Matter....Pages 174-192

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This is the first self-contained summary of non-Noetherian orders in a simple Artinian ring, a subject in which much progress has been made in the last decade.
The contents of the book are mainly Dubrovin valuation rings and semi-hereditary orders, including Pr?fer and semi-local Bezout orders, which are considered, in a sense, as global theories of Dubrovin valuation rings. These are then developed further, and applied to give some examples such as Dubrovin valuation rings in crossed product algebras, semi-hereditary maximal order in certain matrix rings, and the idealizers of semi-hereditary orders and Henselization of Bezout orders.
Audience: This volume will be of interest to researchers and graduate students whose work involves non-commutative ring theory and module theory.
Content:
Front Matter....Pages i-viii
Semi-Hereditary and Pr?fer Orders....Pages 1-20
Dubrovin Valuation Rings....Pages 21-78
Semi-Local Bezout Orders....Pages 79-108
The Applications and Examples....Pages 109-173
Back Matter....Pages 174-192
....