01677nmm a2200277 u 4500001001200000003002700012005001700039007002400056008004100080020001800121050001000139100001600149245006600165246005300231260004800284300002900332653002000361653002100381653002500402041001900427989003200446490003200478856006300510082001000573520081600583EB001852218EBX0100000000000000101652200000000000000.0cr|||||||||||||||||||||181005 ||| eng a9780511542794 4aQA2471 aCohn, P. M.00aFree ideal rings and localization in general ringscP.M. Cohn31aFree Ideal Rings & Localization in General Rings aCambridgebCambridge University Pressc2006 axxii, 572 pagesbdigital aRings (Algebra) aIdeals (Algebra) aAssociative algebras07aeng2ISO 639-2 bCBOaCambridge Books Online0 aNew mathematical monographs40uhttps://doi.org/10.1017/CBO9780511542794xVerlag3Volltext0 a512.4 aProving 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