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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
Namely, a commutative ring $A$ with identity is von Neumann regular if and only if $Ann(I)+P=A$, for every prime ideal $P$ of $A$ and any finitely generated ideal $I$ of $A$ contained in $P$.
📌 3 Key Takeaways
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Objective & Hypothesis:We extend the ring-theoretic concept of going down to algebraic frames and coherent maps.
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Key Findings & Data:We then use the notion introduced to characterize algebraic frames of dimension 0 and frames of dimension at most 1.
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Conclusion & Impact:Namely, a commutative ring $A$ with identity is von Neumann regular if and only if $Ann(I)+P=A$, for every prime ideal $P$ of $A$ and any finitely generated ideal $I$ of $A$ contained in $P$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Algebraic frame · Sublocale · Prime element
Abstract
We extend the ring-theoretic concept of going down to algebraic frames and coherent maps. We then use the notion introduced to characterize algebraic frames of dimension 0 and frames of dimension at most 1. An application to rings yields a characterization of von Neumann regular rings that appears to have hitherto been overlooked. Namely, a commutative ring $A$ with identity is von Neumann regular if and only if $Ann(I)+P=A$, for every prime ideal $P$ of $A$ and any finitely generated ideal $I$ of $A$ contained in $P$.
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Cite This Article
Dube, T. (2022). First steps going down on algebraic frames. Hacettepe Journal of Mathematics and Statistics, Cilt: 48 - Sayı: 6, 1792–1807. https://ajindex.com/articles/529762