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Journals / International Electronic Journal of Algebra / 2025 / Volume: 38 Issue: 38

On the equation $a^{2}x=a$ in unital rings

Grigore Călugăreanu(Babeş-Bolyai University)
Horia F. Pop(Babeş-Bolyai University)
Pages
—
Publication Date
2025-07-14
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Further, we prove results for left strongly regular (square) matrices over various types of rings and propose a conjecture, strongly supported by computational evidence: over commutative rings, left strongly regular matrices are strongly regular.

📌 3 Key Takeaways

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Objective & Hypothesis:In an arbitrary ring, the equation in the title defines the left strongly regular elements.
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Key Findings & Data:While we show that it is not possible to give $2\times 2$ non-examples over $\mathbb{Z}$ or in certain matrix rings over $\mathbb{Z}_{n}$ for $n\in \{8,9,16\}$, we present two examples by George Bergman: a left strongly regular element that is not regular and a regular, left strongly regular element that is not strongly regular.
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Conclusion & Impact:Further, we prove results for left strongly regular (square) matrices over various types of rings and propose a conjecture, strongly supported by computational evidence: over commutative rings, left strongly regular matrices are strongly regular.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Left strongly regular element · strongly regular ring · 2x2 matrix

Abstract

In an arbitrary ring, the equation in the title defines the left strongly regular elements. Elements that are both left and right strongly regular are simply called strongly regular. If all elements of a ring are left strongly regular, then they are, in fact, strongly regular and this is the definition of strongly regular rings. We provide a characterization of when a left strongly regular element is indeed strongly regular, based on an intrinsic condition. While we show that it is not possible to give $2\times 2$ non-examples over $\mathbb{Z}$ or in certain matrix rings over $\mathbb{Z}_{n}$ for $n\in \{8,9,16\}$, we present two examples by George Bergman: a left strongly regular element that is not regular and a regular, left strongly regular element that is not strongly regular. Further, we prove results for left strongly regular (square) matrices over various types of rings and propose a conjecture, strongly supported by computational evidence: over commutative rings, left strongly regular matrices are strongly regular.

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Cite This Article

Călugăreanu, G., Pop, H. F. (2025). On the equation $a^{2}x=a$ in unital rings. International Electronic Journal of Algebra, Volume: 38 Issue: 38. https://doi.org/10.24330/ieja.1578725

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