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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 50 Issue: 5

Structure of rings with commutative factor rings for some ideals contained in their centers

Hai-lan Jin(Yanbian University)0000-0002-0860-9203
Nam Kyun Kim(Hanbat National University)0000-0002-4419-9045
Yang Lee(Daejin University)0000-0002-7572-5191
Zhelin Pıao(Yanbian University)0000-0003-4316-9925
Michal Ziembowski(Warsaw University of Technology)0000-0001-6406-2188
Pages
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Publication Date
2021-10-15
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AI Summary & Key Takeaways

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

We prove that if a non-prime FC ring is noncommutative then it is subdirectly irreducible.

📌 3 Key Takeaways

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Objective & Hypothesis:This article concerns commutative factor rings for ideals contained in the center.
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Key Findings & Data:We prove that (i) for a CIFC ring $R$, $W(R)$ contains all nilpotent elements in $R$ (hence Köthe's conjecture holds for $R$) and $R/W(R)$ is a commutative reduced ring; (ii) $R$ is strongly bounded if $R/N_*(R)$ is commutative and $0\neq N_*(R)\subseteq Z(R)$, where $W(R)$ (resp., $N_*(R)$) is the Wedderburn (resp., prime) radical of $R$.
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Conclusion & Impact:We prove that if a non-prime FC ring is noncommutative then it is subdirectly irreducible.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:CIFC ring · nilradical · center

Abstract

This article concerns commutative factor rings for ideals contained in the center. A ring $R$ is called CIFC if $R/I$ is commutative for some proper ideal $I$ of $R$ with $I\subseteq Z(R)$, where $Z(R)$ is the center of $R$. We prove that (i) for a CIFC ring $R$, $W(R)$ contains all nilpotent elements in $R$ (hence Köthe's conjecture holds for $R$) and $R/W(R)$ is a commutative reduced ring; (ii) $R$ is strongly bounded if $R/N_*(R)$ is commutative and $0\neq N_*(R)\subseteq Z(R)$, where $W(R)$ (resp., $N_*(R)$) is the Wedderburn (resp., prime) radical of $R$. We provide plenty of interesting examples that answer the questions raised in relation to the condition that $R/I$ is commutative and $I\subseteq Z(R)$. In addition, we study the structure of rings whose factor rings modulo nonzero proper ideals are commutative; such rings are called FC. We prove that if a non-prime FC ring is noncommutative then it is subdirectly irreducible.

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Cite This Article

Jin, H., Kim, N. K., Lee, Y., Pıao, Z., Ziembowski, M. (2021). Structure of rings with commutative factor rings for some ideals contained in their centers. Hacettepe Journal of Mathematics and Statistics, Volume: 50 Issue: 5. https://doi.org/10.15672/hujms.729739

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