A Note on Strongly $π$-Regular Elements
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911291035090944 |
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| author | Goyal, Dimple Rani |
| author_facet | Goyal, Dimple Rani |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07305 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Note on Strongly $π$-Regular Elements Goyal, Dimple Rani Rings and Algebras 16E50, 16U99 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. |
| title | A Note on Strongly $π$-Regular Elements |
| topic | Rings and Algebras 16E50, 16U99 |
| url | https://arxiv.org/abs/2502.07305 |