A Note on Strongly $\pi$-Regular Elements

TL;DR

This paper investigates properties of strongly π-regular elements in PI-rings and matrix rings, proving a conjecture about their behavior and providing multiple proofs with different methods.

Contribution

It proves a conjecture on strongly π-regular elements in matrix rings over PI-rings and commutative rings, offering new proofs and structural insights.

Findings

Confirmed the conjecture for matrix rings over commutative rings.

Established the existence of an element Y such that A^n = YA^{n+1}.

Provided multiple independent proofs using different techniques.

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.