Journal of the
Korean Mathematical Society JKMS

Yu showed that every right (left) primitive factor ring of weakly right (left) duo rings is a division ring. It is not difficult to show that each weakly right (left) duo ring is abelian and has the classical right (left) quotient ring. In this note we first provide a left duo ring (but not weakly right duo) in spite of it being left Noetherian and local. Thus we observe conditions under which weakly one-sided duo rings may be two-sided. We prove that a weakly one-sided duo ring $R$ is weakly duo under each of the following conditions: (1) $R$ is semilocal with nil Jacobson radical; (2) $R$ is locally finite. Based on the preceding case (1) we study a kind of composition length of a right or left Artinian weakly duo ring $R$, obtaining that $i(R)$ is finite and $a^{i(R)}R=Ra^{i(R)}=Ra^{i(R)}R$ for all $a\in R$, where $i(R)$ is the index (of nilpotency) of $R$. Note that one-sided Artinian rings and locally finite rings are strongly $\pi$-regular. Thus we also observe connections between strongly $\pi$-regular weakly right duo rings and related rings, constructing available examples.

Keywords: weakly duo ring, duo ring, Artinian ring, Jacobson radical, strongly $\pi$-regular ring, quasi-duo ring

MSC numbers: Primary 16D25, 16L30, 16P20; Secondary 16E50, 16N20, 16N40