Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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RINGS WITH A RIGHT DUO FACTOR RING BY AN IDEAL CONTAINED IN THE CENTER

  • Cheon, Jeoung Soo (Department of Mathematics Pusan National University) ;
  • Kwak, Tai Keun (Department of Mathematics Daejin University) ;
  • Lee, Yang (Department of Mathematics Yanbian University, Institute of Basic Science Daejin University) ;
  • Piao, Zhelin (Department of Mathematics Yanbian University) ;
  • Yun, Sang Jo (Department of Mathematics Dong-A University)
  • Received : 2020.12.04
  • Accepted : 2022.02.10
  • Published : 2022.05.31
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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 first see that this property is seated between right duo and right π-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.

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Acknowledgement

The authors thank the referee for very careful reading of the manuscript and many valuable suggestions that improved the paper by much. This article was supported by the Science and Technology Research Project of Education Department of Jilin Province, China(JJKH20210563KJ).

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