Bulletin of the
Korean Mathematical Society BKMS

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