Bulletin of the
Korean Mathematical Society
BKMS

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

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Bull. Korean Math. Soc.

Published online May 9, 2022

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, and Sang Jo Yun

Pusan National 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 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 fi rst see that this property is seated between right duo and right pi-duo, 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 D3(R) is right CIFD, where D3(R) is the ring of 3 by 3 upper triangular matrices over R whose diagonals are equal. Furthermore, 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 -duo ring, commutative ring

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

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