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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
Finally, we prove that the spectrum $Spec(M)$ is irreducible if and only if $M$ is the finite sum of its submodules, whose $ \mathcal{T}$-radicals are prime in $M$.
📌 3 Key Takeaways
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Objective & Hypothesis:The subject of this paper is the Zariski topology on a multiplication module $M$ over a commutative ring $R$.
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Key Findings & Data:We find a characterization for the radical submodule $rad_{M}(0)$ and also show that there are proper ideals $I_{1},...,I_{n}$ of $R$ such that $rad_{M}(0)=rad_{M}(\left( I_{1}...I_{n}\right) M)$.
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Conclusion & Impact:Finally, we prove that the spectrum $Spec(M)$ is irreducible if and only if $M$ is the finite sum of its submodules, whose $ \mathcal{T}$-radicals are prime in $M$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Multiplication module · prime submodule · spectrum of module
Abstract
The subject of this paper is the Zariski topology on a multiplication module $M$ over a commutative ring $R$. We find a characterization for the radical submodule $rad_{M}(0)$ and also show that there are proper ideals $I_{1},...,I_{n}$ of $R$ such that $rad_{M}(0)=rad_{M}(\left( I_{1}...I_{n}\right) M)$. Finally, we prove that the spectrum $Spec(M)$ is irreducible if and only if $M$ is the finite sum of its submodules, whose $ \mathcal{T}$-radicals are prime in $M$.
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Cite This Article
ÖNEŞ, O., ALKAN, M. (2019). Multiplication modules with prime spectrum. Turkish Journal of Mathematics, Cilt: 43 - Sayı: 4, 2000–2009. https://ajindex.com/articles/208990