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  3. Free Ideal Rings and Localization in General Rings
  • Cited by 92
Publisher:
Cambridge University Press
Online publication date:
August 2009
Print publication year:
2006
Online ISBN:
9780511542794
DOI:
https://doi.org/10.1017/CBO9780511542794
Subjects:
Mathematics (general), Mathematics, Algebra
Series:
New Mathematical Monographs (3)
Information

Book description

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.

Reviews

'This book presents the theory of free ideal rings (firs) in detail.'

Source: L'enseignement mathematique

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