Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2022; 59(3): 529-545

Published online May 31, 2022 https://doi.org/10.4134/BKMS.b201014

Copyright © The Korean Mathematical Society.

Rings with a right duo factor ring by an ideal contained in the center

Jeoung Soo Cheon, Tai Keun Kwak, Yang Lee, Zhelin Piao, Sang Jo Yun

Pusan National University; Daejin University; Daejin University; Yanbian University; Dong-A University

Abstract

This article concerns a ring property that arises from combining one-sided duo factor rings and centers. A ring $R$ is called {\it right CIFD} if $R/I$ is right duo by some proper ideal $I$ of $R$ such that $I$ is contained in the center of $R$. We first see that this property is seated between right duo and right $\pi$-duo\textbf{,} and not left-right symmetric. We prove, for a right CIFD ring $R$, that $W(R)$ coincides with the set of all nilpotent elements of $R$; that $R/P$ is a right duo domain for every minimal prime ideal $P$ of $R$; that $R/W(R)$ is strongly right bounded; and that every prime ideal of $R$ is maximal if and only if $R/W(R)$ is strongly regular, where $W(R)$ is the Wedderburn radical of $R$. It is also proved that a ring $R$ is commutative if and only if $D_3(R)$ is right CIFD, where $D_3(R)$ is the ring of $3$ by $3$ upper triangular matrices over $R$ whose diagonals are equal. Furthermore\textbf{,} we show that the right CIFD property does not pass to polynomial rings, and that the polynomial ring over a ring $R$ is right CIFD if and only if $R/I$ is commutative by a proper ideal $I$ of $R$ contained in the center of $R$.

Keywords: Right duo (factor) ring, right CIFD ring, center, Wedderburn radical, polynomial ring, matrix ring, strongly right bounded ring, right $\pi$-duo ring, commutative ring

MSC numbers: 16D25, 16U70, 16U80, 16N40

Supported by: This article was supported by the Science and Technology Research Project of Education Department of Jilin Province, China(JJKH20210563KJ).