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A STRUCTURE OF NONCENTRAL IDEMPOTENTS

  • Cho, Eun-Kyung (Department of Mathematics Pusan National University) ;
  • Kwak, Tai Keun (Department of Mathematics Daejin University) ;
  • Lee, Yang (Institute of Basic Science Daejin University) ;
  • Piao, Zhelin (Department of Mathematics Pusan National University) ;
  • Seo, Yeon Sook (Department of Mathematics Pusan National University)
  • Received : 2016.10.04
  • Accepted : 2017.07.14
  • Published : 2018.01.31

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.

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

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