Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON RIGHT REGULARITY OF COMMUTATORS

  • Jung, Da Woon (Finance.Fishery.Manufacture Industrial Mathematics Center on Big Data Pusan National University) ;
  • Lee, Chang Ik (Department of Mathematics Pusan National University) ;
  • Lee, Yang (Department of Mathematics Yanbian University and Institute for Applied Mathematics and Optics Hanbat National University) ;
  • Park, Sangwon (Department of Mathematics Dong-A University) ;
  • Ryu, Sung Ju (Department of Mathematics Pusan National University) ;
  • Sung, Hyo Jin (Department of Mathematics Pusan National University)
  • Received : 2021.07.05
  • Accepted : 2021.11.12
  • Published : 2022.07.31
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Abstract

We study the structure of right regular commutators, and call a ring R strongly C-regular if ab - ba ∈ (ab - ba)2R for any a, b ∈ R. We first prove that a noncommutative strongly C-regular domain is a division algebra generated by all commutators; and that a ring (possibly without identity) is strongly C-regular if and only if it is Abelian C-regular (from which we infer that strong C-regularity is left-right symmetric). It is proved that for a strongly C-regular ring R, (i) if R/W(R) is commutative, then R is commutative; and (ii) every prime factor ring of R is either a commutative domain or a noncommutative division ring, where W(R) is the Wedderburn radical of R.

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Acknowledgement

The authors thank the referee for a very careful reading of the manuscript and many valuable suggestions that greatly improved the paper. This study was supported by research funds from Dong-A University.

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