Free Ideal Rings and Localization in General Rings
Seiten
2006
Cambridge University Press (Verlag)
978-0-521-85337-8 (ISBN)
Cambridge University Press (Verlag)
978-0-521-85337-8 (ISBN)
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, and there is also a full account of localization. Each chapter has a number of exercises plus open problems and historical notes.
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.
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.
Paul Cohn is a Emeritus Professor of Mathematics at the University of London and Honorary Research Fellow at University College London.
Preface; 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; 3. Factorization; 4. 2-firs with a distributive factor lattice; 5. Modules over firs and semifirs; 6. Centralizers and subalgebras; 7. Skew fields of fractions; Appendix; Bibliography and author index; Subject index.
| Erscheint lt. Verlag | 8.6.2006 |
|---|---|
| Reihe/Serie | New Mathematical Monographs |
| Zusatzinfo | Worked examples or Exercises; 38 Line drawings, unspecified |
| Verlagsort | Cambridge |
| Sprache | englisch |
| Maße | 160 x 234 mm |
| Gewicht | 961 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| ISBN-10 | 0-521-85337-0 / 0521853370 |
| ISBN-13 | 978-0-521-85337-8 / 9780521853378 |
| Zustand | Neuware |
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