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Proving that a polynomial ring in one variable over a field is a principal ideal domain can be done by means of the Euclidean algorithm, but this does not extend to more variables. However, if the variables are not allowed to commute, giving a free associative algebra, then there is a generalization, the weak algorithm, which can be used to prove that all one-sided ideals are free. This book presents the theory of free ideal rings (firs) in detail. Particular emphasis is placed on rings with a weak algorithm, exemplified by free associative algebras. There is also a full account of localization which is treated for general rings but the features arising in firs are given special attention. Each section has a number of exercises, including some open problems, and each chapter ends in a historical note.Read more
- This theory not found in any other book
- Subject is smoothly developed and well motivated
- Noncommutative theory has relations to many other topics
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'This book presents the theory of free ideal rings (firs) in detail.' L'enseignement mathematique
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- Date Published: June 2006
- format: Hardback
- isbn: 9780521853378
- length: 594 pages
- dimensions: 234 x 160 x 34 mm
- weight: 0.961kg
- contains: 38 b/w illus. 864 exercises
- availability: Temporarily unavailable - available from TBC
Table of Contents
Note to the reader
Terminology, notations and conventions used
List of special notation
0. Preliminaries on modules
1. Principal ideal domains
2. Firs, semifirs and the weak algorithm
4. 2-firs with a distributive factor lattice
5. Modules over firs and semifirs
6. Centralizers and subalgebras
7. Skew fields of fractions
Bibliography and author index
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