Rings in which one-sided strongly $π$-regular elements are strongly $π$-regular
Dimple Rani Goyal, Dinesh Khurana
TL;DR
The paper investigates whether one-sided strongly π-regular elements are strongly π-regular in Dedekind-finite rings beyond exchange rings, addressing Hartwig and Luh's question. It develops the Dischinger framework (right vs left) and shows the property is not symmetric. It proves broad sufficient conditions for a ring to be right Dischinger, including when nilpotent elements form nil subrings, rings with bounded index of nilpotence, Noetherian or Goldie-type rings, and various one-sided classes like weakly semicommutative, one-sided duo, and linearly McCoy. It also presents negative examples illustrating one-sided asymmetry and discusses open questions. The results extend earlier work by Azumaya and Dittmer et al., bridging the gap between one-sided π-regularity and global strong π-regularity in Dedekind-finite contexts.
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.