Bulletin of the
Korean Mathematical Society
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ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2018; 55(1): 25-40

Published online January 31, 2018 https://doi.org/10.4134/BKMS.b160801

Copyright © The Korean Mathematical Society.

A structure of noncentral idempotents

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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