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

78106

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

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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Hacettepe Journal of Mathematics and Statistics-Cover
  • Yayın Aralığı: 6
  • Başlangıç: 2002
  • Yayıncı: Hacettepe Üniversitesi Fen Fakultesi
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