A Note on Strongly $\pi$-Regular Elements

Abstract

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.