WEAKLY DUO RINGS WITH NIL JACOBSON RADICAL

Abstract

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 <TEX>$\alpha^{i(R)}R\;=\;R\alpha^{i(R)\;=\;R\alpha^{i(R)}R\;for\;all\;\alpha\;{\in}\;R$</TEX>, where i(R) is the index (of nilpotency) of R. Note that one-sided Artinian rings and locally finite rings are strongly <TEX>$\pi-regular$</TEX>. Thus we also observe connections between strongly <TEX>$\pi-regular$</TEX> weakly right duo rings and related rings, constructing available examples.