A structure of noncentral idempotents
Bull. Korean Math. Soc. 2018 Vol. 55, No. 1, 25-40
Published online January 31, 2018
Eun-Kyung Cho, Tai Keun Kwak, Yang Lee, Zhelin Piao, Yeon Sook Seo
Pusan National University, Daejin University, Daejin University, Yanbian University, Pusan National University
Abstract : We focus on the structure of the set of noncentral idempotents whose role is similar to one of central idempotents. We introduce the concept of quasi-Abelian rings which unit-regular rings satisfy. We first observe that the class of quasi-Abelian rings is seated between Abelian and direct finiteness. It is proved that a regular ring is directly finite if and only if it is quasi-Abelian. It is also shown that quasi-Abelian property is not left-right symmetric, but left-right symmetric when a given ring has an involution. Quasi-Abelian property is shown to do not pass to polynomial rings, comparing with Abelian property passing to polynomial rings.
Keywords : right quasi-Abelian ring, idempotent, Abelian ring, semisimple Artinian ring, directly finite ring, group of units
MSC numbers : 16U80, 16S36, 16W10
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