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An important object associated to a ring is the group of its units. Groups of units are interesting for the following branches of mathematics: group theory, ring theory, ktheory, representation theory, the theory of group rings, topology, the algebraic theory of numbers. In general topology there exists various natural generalizations of compact spaces: countably compact, pseudocompact, ?compact and sequentially compact. These classes of topological groups are interesting objects of study for the theory of topological groups. Evidently, their topology are more difficult in comparison with compact groups, therefore that groups inherit some nice properties of compact groups. In this book we study the groups of units of contably compact, pseudocompact, totally bounded and more general, linearly compact rings. The topology of compact rings is relatively well studied. From this point of view the topological structure of totally bounded rings is richer and diverse. Since the completion of a totally bounded ring is compact, the methods of compact rings can be applied in the study of totally bounded rings.
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