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Journals / Turkish Journal of Mathematics / 2014 / Cilt: 38 - Sayı: 4

On density theorems for rings of Krull type with zero divisors

Başak SAYLAM AY
Pages
614–624
Publication Date
2022-11-03
DOI
—
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AI Summary & Key Takeaways

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

We prove that there is an order homomorphism from I(R) into the cardinal direct sum \coprodi \in I Gi and we investigate the conditions that make this monomorphism onto for R.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Let R be a commutative ring and I(R) denote the multiplicative group of all invertible fractional ideals of R, ordered by A \leqslant B if and only if B \subseteq A.
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Key Findings & Data:If R is a Marot ring of Krull type, then R(Pi), where {Pi}i \in I are a collection of prime regular ideals of R, is a valuation ring and R = \bigcap R(Pi).
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Conclusion & Impact:We prove that there is an order homomorphism from I(R) into the cardinal direct sum \coprodi \in I Gi and we investigate the conditions that make this monomorphism onto for R.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Krull ring · ring of Krull type · valuation Marot ring

Abstract

Let R be a commutative ring and I(R) denote the multiplicative group of all invertible fractional ideals of R, ordered by A \leqslant B if and only if B \subseteq A. If R is a Marot ring of Krull type, then R(Pi), where {Pi}i \in I are a collection of prime regular ideals of R, is a valuation ring and R = \bigcap R(Pi). We denote by Gi the value group of the valuation associated with R(Pi). We prove that there is an order homomorphism from I(R) into the cardinal direct sum \coprodi \in I Gi and we investigate the conditions that make this monomorphism onto for R.

Özet

Let R be a commutative ring and I(R) denote the multiplicative group of all invertible fractional ideals of R, ordered by A \leqslant B if and only if B \subseteq A. If R is a Marot ring of Krull type, then R(Pi), where {Pi}i \in I are a collection of prime regular ideals of R, is a valuation ring and R = \bigcap R(Pi). We denote by Gi the value group of the valuation associated with R(Pi). We prove that there is an order homomorphism from I(R) into the cardinal direct sum \coprodi \in I Gi and we investigate the conditions that make this monomorphism onto for R.

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Cite This Article

AY, B. S. (2022). On density theorems for rings of Krull type with zero divisors. Turkish Journal of Mathematics, Cilt: 38 - Sayı: 4, 614–624. https://ajindex.com/articles/608259

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References (6)

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