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

Year 2021, Volume: 50 Issue: 5, 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

References

• [1] G. Birkhoff, Subdirect unions in universal algebra, Bull. Amer. Math. Soc. 50, 764– 768, 1944.
• [2] C. Faith, Algebra II, Ring theory, Springer-Verlag, Berlin-New York, 1976.
• [3] E.H. Feller, Properties of primary noncommutative rings, Trans. Amer. Math. Soc. 89, 79–91, 1958.
• [4] C. Huh, S.H. Jang, C.O. Kim and Y. Lee, Rings whose maximal one-sided ideals are two-sided, Bull. Korean Math. Soc. 39, 411–422, 2002.
• [5] C. Huh, H.K. Kim and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167, 37–52, 2002.
• [6] S.U. Hwang, Y.C. Jeon and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302, 186–199, 2006.
• [7] N. Jacobson, The theory of rings, American Mathematical Society Mathematical Surveys II, American Mathematical Society, New York, 1943.
• [8] Y.C. Jeon, H.K. Kim, Y. Lee and J.S. Yoon, On weak Armendariz rings, Bull. Korean Math. Soc. 46, 135–146, 2009.
• [9] N.K. Kim, T.K. Kwak and Y. Lee, On a generalization of right duo rings, Bull. Korean Math. Soc. 53, 925–942, 2016.
• [10] A. Leroy, J. Matczuk and E.R. Puczylowski, Quasi-duo skew polynomial rings, J. Pure Appl. Algebra 212, 1951–1959, 2008.
• [11] H.-P. Yu, On quasi-duo rings, Glasgow Math. J. 37, 21–31, 1995.