Structure of rings with commutative factor rings for some ideals contained in their centers
| Autor: | Hai-lan Jin, Nam Kyun Kim, Michał Ziembowski, Zhelin Piao, Yang Lee |
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| Rok vydání: | 2021 |
| Předmět: |
Statistics and Probability
Matematik Pure mathematics Algebra and Number Theory Mathematics::Commutative Algebra Structure (category theory) Factor (chord) Simple ring CIFC ring nilradical center strongly bounded ring right quasi-duo ring FC ring simple ring non-prime FC ring Center (algebra and category theory) Geometry and Topology Commutative property Mathematics Analysis |
| Zdroj: | Volume: 50, Issue: 5 1280-1291 Hacettepe Journal of Mathematics and Statistics |
| ISSN: | 2651-477X |
| DOI: | 10.15672/hujms.729739 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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