Anéis unicamente limpos e anéis de grupo unicamente limpos.

Abstract

Units and idempotents are key elements in determining the structure of a ring. In particular, Peirce’s decomposition induced by an idempotent of a ring helps us to define and classify new types of rings. A ring is said to be clean if all its elements can be written as the sum of an idempotent and a unit. This notion was motivated by the study of exchange rings by Nicholson in 1977 and is closely connected with some other important notions of rings. A ring is called uniquely clean if the previous decomposition is unique. Such a class of rings was first studied by Anderson and Camillo in 2002 for the commutative case. The first part of this work is devoted to the study of non-commutative uniquely clean rings by Nicholson and Zhou in 2004, where they proved that the structure of such rings is very close to that of Boolean rings. Group ring theory plays a central role in the development of group representation theory and attracts researchers from other branches of mathematics such as homology algebra and algebraic topology. The second part of the dissertation seeks to study under what conditions a group ring is uniquely clean. Such a question was answered by Chen and Nicholson in 2006. Nonetheless, a number of issues involving clean rings and uniquely clean rings remain open.