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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
We thereby generalize Domsha, Vasiunyk, and Zabavsky's theorems to a much wider class of rings.
📌 3 Key Takeaways
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Objective & Hypothesis:We combine Euclidean and adequate rings, and introduce a new type of ring.
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Key Findings & Data:A ring $R$ is called an E-adequate ring provided that for any $a,b\in R$ such that $aR+bR=R$ and $c\neq 0$ there exists $y\in R$ such that $(a+by,c)$ is an E-adequate pair.
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Conclusion & Impact:We thereby generalize Domsha, Vasiunyk, and Zabavsky's theorems to a much wider class of rings.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Euclidean rings · adequate rings · elementary divisor rings
Abstract
We combine Euclidean and adequate rings, and introduce a new type of ring. A ring $R$ is called an E-adequate ring provided that for any $a,b\in R$ such that $aR+bR=R$ and $c\neq 0$ there exists $y\in R$ such that $(a+by,c)$ is an E-adequate pair. We shall prove that an E-adequate ring is an elementary divisor ring if and only if it is a Hermite ring. Elementary matrix reduction over such rings is also studied. We thereby generalize Domsha, Vasiunyk, and Zabavsky's theorems to a much wider class of rings.
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Cite This Article
CHEN, H., SHEIBANI, M. (2022). Combining Euclidean and adequate rings. Turkish Journal of Mathematics, Cilt: 40 - Sayı: 3, 506–516. https://ajindex.com/articles/608086