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

Authors : Hailan JİN, Nam Kyun KİM, Yang LEE, Zhelin PIAO, Michal ZİEMBOWSKİ

Pages : 1280-1291

Doi:10.15672/hujms.729739

View : 24 | Download : 11

Publication Date : 2021-10-15

Article Type : Research Paper

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 $I\subseteq Zinsert ignore into journalissuearticles values(R);$, where $Zinsert ignore into journalissuearticles values(R);$ is the center of $R$. We prove that insert ignore into journalissuearticles values(i); for a CIFC ring $R$, $Winsert ignore into journalissuearticles values(R);$ contains all nilpotent elements in $R$ insert ignore into journalissuearticles values(hence Köthe`s conjecture holds for $R$); and $R/Winsert ignore into journalissuearticles values(R);$ is a commutative reduced ring; insert ignore into journalissuearticles values(ii); $R$ is strongly bounded if $R/N_*insert ignore into journalissuearticles values(R);$ is commutative and $0\neq N_*insert ignore into journalissuearticles values(R);\subseteq Zinsert ignore into journalissuearticles values(R);$, where $Winsert ignore into journalissuearticles values(R);$ insert ignore into journalissuearticles values(resp., $N_*insert ignore into journalissuearticles values(R);$); is the Wedderburn insert ignore into journalissuearticles values(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 $I\subseteq Zinsert ignore into journalissuearticles values(R);$. 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