arXiv subjects

Dimple Rani Goyal

Publications and source records attributed to Dimple Rani Goyal.

2 recordsLinked to original sources

arXiv2025

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

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.

math.RA↗

arXiv2025

A Note on Strongly $\pi$-Regular Elements

In this article, we prove that in a PI-ring (or polynomial identity ring) $S$, for an element $A \in \mathbb{M}_m(S)$ if $A^n= A^{n+1}X$ for some $n \in \mathbb{N}$ and $X \in \mathbb{M}_m(S)$, then there exists an element $Y\in \mathbb{M}_m(S)$ such that $A^n = YA^{n+1}$. As a consequence, we show that this property also holds in matrix rings over commutative rings, thereby confirming a recent conjecture proposed by C\u{a}lug\u{a}reanu and Pop. Moreover, we present another independent proofs of this conjecture, highlighting different structural approaches and techniques.

math.RA↗