A Note on Strongly $\ddot{I}€$-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 $A^n=Y{A}^{n+1}$. As a consequence, we show that this property also holds in matrix rings over commutative rings, thereby confirming a recent conjecture proposed by Călugăreanu and Pop. Moreover, we present another independent proofs of this conjecture, highlighting different structural approaches and techniques.