Free Ideal Rings and Localization in General Rings

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.