• 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


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.


Supported by : National Research Foundation of Korea(NRF)


  1. G. Ehrlich, Unit-regular rings, Portugal. Math. 27 (1968), 209-212.
  2. K. R. Goodearl, Von Neumann Regular Rings, Pitman, London, 1979.
  3. S. U. Hwang, Y. C. Jeon, and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302 (2006), no. 1, 186-199.
  4. N. Jacobson, Some remarks on one-sided inverses, Proc. Amer. Math. Soc. 1 (1950), 352-355.
  5. N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488.
  6. J. V. Neumann, On regular rings, Proceedngs of the National Academy of Sci. 22 (1936), 707-713.
  7. J. C. Shepherdson, Inverses and zero-divisors in matrix ring, Proc. London Math. Soc. 1 (1951), 71-85.