Rings in which one-sided strongly $\pi$-regular elements are strongly $\pi$-regular

TL;DR

This paper investigates the properties of rings where one-sided strongly π-regular elements are also strongly π-regular, extending previous results and examining the symmetry of these conditions across different classes of Dedekind-finite rings.

Contribution

The paper generalizes the understanding of strongly π-regular elements in various Dedekind-finite rings and demonstrates the non-symmetry of certain conditions.

Findings

The condition is not left-right symmetric.

Extension of results to various classes of Dedekind-finite rings.

Clarification of the relationship between one-sided and two-sided strongly π-regular elements.

Abstract

In 1977, Hartwig and Luh asked if $a$ is an element of a Dedekind-finite ring $S$, then does $aS = a^2S$ imply $Sa = Sa^2$. 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.