Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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STRUCTURE OF UNIT-IFP RINGS

  • Lee, Yang (Institute of Basic Science Daejin University)
  • Received : 2017.10.22
  • Accepted : 2018.04.09
  • Published : 2018.09.01
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Abstract

In this article we first investigate a sort of unit-IFP ring by which Antoine provides very useful information to ring theory in relation with the structure of coefficients of zero-dividing polynomials. Here we are concerned with the whole shape of units and nilpotent elements in such rings. Next we study the properties of unit-IFP rings through group actions of units on nonzero nilpotent elements. We prove that if R is a unit-IFP ring such that there are finite number of orbits under the left (resp., right) action of units on nonzero nilpotent elements, then R satisfies the descending chain condition for nil left (resp., right) ideals of R and the upper nilradical of R is nilpotent.

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References

  1. S. A. Amitsur, Radicals of polynomial rings, Canad. J. Math. 8 (1956), 355-361. https://doi.org/10.4153/CJM-1956-040-9
  2. R. Antoine, Nilpotent elements and Armendariz rings, J. Algebra 319 (2008), no. 8, 3128-3140. https://doi.org/10.1016/j.jalgebra.2008.01.019
  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. G. M. Bergman, Modules over coproducts of rings, Trans. Amer. Math. Soc. 200 (1974), 1-32. https://doi.org/10.1090/S0002-9947-1974-0357502-5
  5. J. Han and S. Park, Rings with a finite number of orbits under the regular action, J. Korean Math. Soc. 51 (2014), no. 4, 655-663. https://doi.org/10.4134/JKMS.2014.51.4.655
  6. C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), no. 2, 751-761. https://doi.org/10.1081/AGB-120013179
  7. H. K. Kim, T. K. Kwak, Y. Lee, and Y. Seo, Insertion of units at zero products, J. Algebra Appl. 17 (2018), no. 3, 1850043, 20 pp. https://doi.org/10.1142/S0219498818500433
  8. 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
  9. G. Marks, On 2-primal Ore extensions, Comm. Algebra 29 (2001), no. 5, 2113-2123. https://doi.org/10.1081/AGB-100002173