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

Year 2021, , 1280 - 1291, 15.10.2021

Hai-lan Jin Nam Kyun Kim , Yang Lee , Zhelin Pıao , Michal Ziembowski

https://doi.org/10.15672/hujms.729739

Cited By: 1

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.

Keywords

CIFC ring, nilradical, center, strongly bounded ring, right quasi-duo ring, FC ring, simple ring, non-prime FC ring

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