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ON REVERSIBILITY RELATED TO IDEMPOTENTS

  • Jung, Da Woon (Finance Fishery Manufacture Industrial Mathematics Center on Big Data Pusan National University) ;
  • Lee, Chang Ik (Department of Mathematics Pusan National University) ;
  • Lee, Yang (Department of Mathematics Yanbian University) ;
  • Park, Sangwon (Department of Mathematics Dong-A University) ;
  • Ryu, Sung Ju (Department of Mathematics Pusan National University) ;
  • Sung, Hyo Jin (Department of Mathematics Pusan National University) ;
  • Yun, Sang Jo (Department of Mathematics Dong-A University)
  • Received : 2018.08.14
  • Accepted : 2018.11.21
  • Published : 2019.07.31

Abstract

This article concerns a ring property which preserves the reversibility of elements at nonzero idempotents. A ring R shall be said to be quasi-reversible if $0{\neq}ab{\in}I(R)$ for a, $b{\in}R$ implies $ba{\in}I(R)$, where I(R) is the set of all idempotents in R. We investigate the quasi-reversibility of 2 by 2 full and upper triangular matrix rings over various kinds of reversible rings, concluding that the quasi-reversibility is a proper generalization of the reversibility. It is shown that the quasi-reversibility does not pass to polynomial rings. The structure of Abelian rings is also observed in relation with reversibility and quasi-reversibility.

Keywords

References

  1. D. D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra 27 (1999), no. 6, 2847-2852. https://doi.org/10.1080/00927879908826596
  2. A. Badawi, On abelian ${\pi}$-regular rings, Comm. Algebra 25 (1997), no. 4, 1009-1021. https://doi.org/10.1080/00927879708825906
  3. H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2 (1970), 363-368. https://doi.org/10.1017/S0004972700042052
  4. P. M. Cohn, Reversible rings, Bull. London Math. Soc. 31 (1999), no. 6, 641-648. https://doi.org/10.1112/S0024609399006116
  5. K. R. Goodearl, von Neumann Regular Rings, Monographs and Studies in Mathematics, 4, Pitman (Advanced Publishing Program), Boston, MA, 1979.
  6. 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
  7. S. U. Hwang, Y. C. Jeon, and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302 (2006), no. 1, 186-199. https://doi.org/10.1016/j.jalgebra.2006.02.032
  8. D. W. Jung, N. K. Kim, Y. Lee, and S. J. Ryu, On properties related to reversible rings, Bull. Korean Math. Soc. 52 (2015), no. 1, 247-261. https://doi.org/10.4134/BKMS.2015.52.1.247
  9. N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488. https://doi.org/10.1006/jabr.1999.8017
  10. N. K. Kim and Y. Lee, On right quasi-duo rings which are ${\pi}$-regular, Bull. Korean Math. Soc. 37 (2000), no. 2, 217-227.
  11. N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 207-223. https://doi.org/10.1016/S0022-4049(03)00109-9
  12. T. K. Kwak, S. I. Lee, and Y. Lee, Quasi-normality of idempotents on nilpotents, Hacettepe J. Math. Stat. (to appear).
  13. G. Marks, On 2-primal Ore extensions, Comm. Algebra 29 (2001), no. 5, 2113-2123. https://doi.org/10.1081/AGB-100002173
  14. 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
  15. J. von Neumann, On regular rings, Proceedngs of the National Academy of Sciences 22 (1936), 707-713. https://doi.org/10.1073/pnas.22.12.707
  16. A. Smoktunowicz, Polynomial rings over nil rings need not be nil, J. Algebra 233 (2000), no. 2, 427-436. https://doi.org/10.1006/jabr.2000.8451