Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Special Right Jacobson Radicals for Right Near-rings

  • Received : 2011.12.26
  • Accepted : 2012.12.18
  • Published : 2014.12.23
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Abstract

In this paper three more right Jacobson-type radicals, $J^r_{g{\nu}}$, are introduced for near-rings which generalize the Jacobson radical of rings, ${\nu}{\in}\{0,1,2\}$. It is proved that $J^r_{g{\nu}}$ is a special radical in the class of all near-rings. Unlike the known right Jacobson semisimple near-rings, a $J^r_{g{\nu}}$-semisimple near-ring R with DCC on right ideals is a direct sum of minimal right ideals which are right R-groups of type-$g_{\nu}$, ${\nu}{\in}\{0,1,2\}$. Moreover, a finite right $g_2$-primitive near-ring R with eRe a non-ring is a near-ring of matrices over a near-field (which is isomorphic to eRe), where e is a right $g_2$-primitive idempotent in R.

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References

  1. G. L. Booth and N. J. Groenewald, Special radicals of near-rings, Math. Japonica, 37(4)(1992), 701-706.
  2. G. L. Booth, N. J. Groenewald, and S. Veldsman, A Kurosh-Amitsur prime radical for near-rings, Comm. Algebra, 18(9)(1990), 3111-3122. https://doi.org/10.1080/00927879008824063
  3. K. Kaarli, On Jacobson-type radicals of near-rings, Acta Math. Hungar., 50(1987), 71-78. https://doi.org/10.1007/BF01903365
  4. G. Pilz, Near-rings, revised ed., North-Holland, Amsterdam, 1983.
  5. R. Srinivasa Rao and K. Siva Prasad, Hereditary right Jacobson radicals of type-1(e) and 2(e) for right near-rings, submitted for publication.
  6. R. Srinivasa Rao and K. Siva Prasad, A radical for right near-rings: The right Jacobson radical of type-0, Int. J. Math. Math. Sci., 2006(16)(2006), 1-13, Article ID 68595.
  7. R. Srinivasa Rao and K. Siva Prasad, Two more radicals for right near-rings: The right Jacobson radicals of type-1 and 2, Kyungpook Math. J., 46(4)(2006), 603-613.
  8. R. Srinivasa Rao and K. Siva Prasad, Right semisimple right near-rings, Southeast Asian Bull. Math., 33(6)(2009), 1189-1205.
  9. R. Srinivasa Rao, K. Siva Prasad, and T. Sinivas, Kurosh-Amitsur right Jacobson radical of type-0 for right near-rings, Int. J. Math. Math. Sci., 2008(1)(2008), 1-6, Article ID 741609.
  10. R. Srinivasa Rao, K. Siva Prasad, and T. Sinivas, Kurosh-Amitsur right Jacobson radicals of type-1 and 2 for right near-rings, Result. Math., 51(3-4)(2008), 309-317. https://doi.org/10.1007/s00025-007-0280-2
  11. R. Srinivasa Rao, K. Siva Prasad, and T. Sinivas, Hereditary right Jacobson radical of type-0(e) for right near-rings, Beitr. Algebra Geom., 50(1)(2009), 11-23.
  12. S. Veldsman, Modulo-constant ideal-hereditary radicals of near-rings, Quaest. Math., 11(3)(1988), 253-278. https://doi.org/10.1080/16073606.1988.9632143

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  1. A module theoretic characterization of the prime radical of near-rings 2018, https://doi.org/10.1007/s13366-017-0349-3
  2. Right representations of right near-ring radicals pp.2190-7668, 2019, https://doi.org/10.1007/s13370-018-0627-8