Bulletin of the
Korean Mathematical Society
BKMS

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

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Bull. Korean Math. Soc. 2022; 59(4): 853-868

Published online July 31, 2022 https://doi.org/10.4134/BKMS.b210495

Copyright © The Korean Mathematical Society.

On right regularity of commutators

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

Pusan National University; Pusan National University; Hanbat National University; Dong-A University; Pusan National University; Pusan National 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

Supported by: This study was supported by research funds from Dong-A University.