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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
We show that $R\{X\}$ is a Prüfer ring if and only if $R$ is a von Neumann regular ring, and so if $R\{X\}$ satisfies one of the Prüfer conditions, it satisfies all of them.
📌 3 Key Takeaways
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Objective & Hypothesis:Let $R$ be a commutative ring with unity and $W=\{f(X)\in R[X]:f(0)=1\}$.
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Key Findings & Data:We define $R\{X\}=W^{-1}R[X]$.
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Conclusion & Impact:We show that $R\{X\}$ is a Prüfer ring if and only if $R$ is a von Neumann regular ring, and so if $R\{X\}$ satisfies one of the Prüfer conditions, it satisfies all of them.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Prime ideal · localization · over ring
Abstract
Let $R$ be a commutative ring with unity and $W=\{f(X)\in R[X]:f(0)=1\}$. We define $R\{X\}=W^{-1}R[X]$. We show that the maximal ideals of $R\{X\} $ are of the form $W^{-1}(M,X)$ where $M$ is a maximal ideal of $R$, and so if $R$ is finite dimensional, then $\dim R\{X\}=\dim R[X]$. We show that $R\{X\}$ is a Prüfer ring if and only if $R$ is a von Neumann regular ring, and so if $R\{X\}$ satisfies one of the Prüfer conditions, it satisfies all of them.
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Cite This Article
Abuosba, E., Al-azaizeh, M. (2025). The ring $R\{X\}$. International Electronic Journal of Algebra, Volume: 37 Issue: 37. https://doi.org/10.24330/ieja.1478925