Rings in which one-sided strongly $\ddot{I}€$-regular elements are strongly $\ddot{I}€$-regular

Abstract

In 1977, Hartwig and Luh asked if $a$ is an element of a Dedekind-finite ring $S$, then does $aS=a^{2}S$ imply $Sa=S{a}^2$. This question was answered negatively by Dittmer, Khurana, and Nielsen in 2014. On the other hand, Dittmer et al. proved that the question of Hartwig and Luh has a positive answer for Dedekind-finite exchange rings. We explore the question of Hartwig and Luh for various other classes of Dedekind-finite rings. We will also prove that the condition in question is not left-right symmetric.