A Note on Strongly $π$-Regular Elements

TL;DR

The article investigates when left-right strong π-regularity lifts from a base ring to matrix rings. It proves that for $A \in M_m(S)$ with $S$ a PI-ring, the condition $A^n\,\mathbb{M}_m(S) = A^{n+1}\,\mathbb{M}_m(S)$ forces $\mathbb{M}_m(S)A^n = \mathbb{M}_m(S)A^{n+1}$, making $A$ strongly π-regular on both sides. Consequently, for commutative $S$, the CalPop conjecture holds, and the paper provides three independent proofs of this corollary (via determinant identities, exchange-ring embeddings, and prime-factor arguments). An illustrative Shepherdson-type example demonstrates that local-to-global lifting to matrix rings is not guaranteed in general. The work highlights multiple structural approaches to strongly π-regularity in matrix rings, with implications for Morita invariance and PI-ring theory.

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 Călugăreanu and Pop. Moreover, we present another independent proofs of this conjecture, highlighting different structural approaches and techniques.

Theorems & Definitions (16)

• Lemma 1
• Proposition 2
• proof
• Theorem 3
• Theorem 4
• Corollary 5
• Lemma 6
• Corollary 7
• Theorem 8
• Theorem 9