# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online May 12, 2022

## On right regularity of commutators

Da Woon Jung, Chang Ik Lee, Yang Lee, Sangwon Park, Sung Ju Ryu, and Hyo Jin Sung

Center of Big Data, Pusan National University, Yanbian University, Dong-A University

### Abstract

We study the structure of right regular commutators, and call a ring $R$ {\it strongly C-regular} if $ab-ba\in (ab-ba)^2R$ for any $a, b\in R$. 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 $R$.

Keywords: commutator, strongly C-regular ring, right regular, commutator ideal, Abelian ring, division ring, nilradical, Jacobson radical, prime factor ring, center

MSC numbers: 16E50, 16U80, 16N40, 16N20, 16N60