The subject matter of this book lies at the crossroads of four areas: ring theory, group theory, group representation theory, and algebraic number theory. Like so many interdisciplinary studies, it has its fascinations and attractions, but also its inherent dilemmas. A multitude of concepts and devices pertaining to the structure of groups and rings, algebraic АГ-theory, algebraic and analytic number theory, valuation theory, and group representation theory are used in open or disguised form to investigate the structure of the unit group of a particular ring. This provides for a vivid interplay among, and is a source of enrichment for, the mentioned disciplines. The leitmotif in our discussion is created by the following two related problems.