DOI QR코드

DOI QR Code

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

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.

Keywords

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).

References

  1. A. Badawi, On abelian π-regular rings, Comm. Algebra 25 (1997), no. 4, 1009-1021. https://doi.org/10.1080/00927879708825906
  2. A. W. Chatters and W. M. Xue, On right duo p.p. rings, Glasgow Math. J. 32 (1990), no. 2, 221-225. https://doi.org/10.1017/S0017089500009253
  3. H. Chen, Y. Lee, and Z. Piao, Structures related to right duo factor rings, Kyungpook Math. J. 61 (2021), no. 1, 11-21. https://doi.org/10.5666/KMJ.2021.61.1.11
  4. K.-J. Choi, T. K. Kwak, and Y. Lee, Reversibility and symmetry over centers, J. Korean Math. Soc. 56 (2019), no. 3, 723-738. https://doi.org/10.4134/JKMS.j180364
  5. C. Faith, Algebra. II, Grundlehren der Mathematischen Wissenschaften, No. 191, Springer-Verlag, Berlin, 1976.
  6. E. H. Feller, Properties of primary noncommutative rings, Trans. Amer. Math. Soc. 89 (1958), 79-91. https://doi.org/10.2307/1993133
  7. K. R. Goodearl, von Neumann regular rings, Monographs and Studies in Mathematics, 4, Pitman (Advanced Publishing Program), Boston, MA, 1979.
  8. I. N. Herstein, Topics in Ring Theory, The University of Chicago Press, Chicago, IL, 1969.
  9. Y. Hirano, C. Hong, J. Kim, and J. K. Park, On strongly bounded rings and duo rings, Comm. Algebra 23 (1995), no. 6, 2199-2214. https://doi.org/10.1080/00927879508825341
  10. C. Y. Hong, H. K. Kim, N. K. Kim, T. K. Kwak, and Y. Lee, One-sided duo property on nilpotents, Hacet. J. Math. Stat. 49 (2020), no. 6, 1974-1987. https://doi.org/10.15672/hujms.571016
  11. C. Huh, H. K. Kim, and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167 (2002), no. 1, 37-52. https://doi.org/10.1016/S0022-4049(01)00149-9
  12. H. Jin, N. K. Kim, Y. Lee, Z. Piao, and M. Ziembowski, Structure of rings with commutative factor rings for some ideals contained in their centers, Hacet. J. Math. Stat. 50 (2021), no. 5, 1280-1291. https://doi.org/10.15672/hujms.729739
  13. N. K. Kim, T. K. Kwak, and Y. Lee, On a generalization of right duo rings, Bull. Korean Math. Soc. 53 (2016), no. 3, 925-942. https://doi.org/10.4134/BKMS.b150441
  14. J. Lambek, Lectures on Rings and Modules, Blaisdell Publishing Co., Waltham, MA, 1966.
  15. Y. Lee, On generalizations of commutativity, Comm. Algebra 43 (2015), no. 4, 1687-1697. https://doi.org/10.1080/00927872.2013.876035
  16. N. H. McCoy, Generalized regular rings, Bull. Amer. Math. Soc. 45 (1939), no. 2, 175-178. https://doi.org/10.1090/S0002-9904-1939-06933-4
  17. M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17. http://projecteuclid.org/euclid.pja/1195510144 https://doi.org/10.3792/pjaa.73.14
  18. G. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43-60. https://doi.org/10.2307/1996398