%0 eBook
%M Solr-EB001852218
%A Cohn, P. M.
%I Cambridge University Press
%D 2006
%C Cambridge
%G English
%B New mathematical monographs
%@ 9780511542794
%T Free ideal rings and localization in general rings
%U https://doi.org/10.1017/CBO9780511542794
%X 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