Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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WEAKLY DUO RINGS WITH NIL JACOBSON RADICAL

  • KIM HONG KEE (Department of Mathematics Gyeongsang National University) ;
  • KIM NAM KYUN (Division of General Education Hanbat National University) ;
  • LEE YANG (Department of Mathematics Education Pusan National University)
  • Published : 2005.05.01
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Abstract

Yu showed that every right (left) primitive factor ring of weakly right (left) duo rings is a division ring. It is not difficult to show that each weakly right (left) duo ring is abelian and has the classical right (left) quotient ring. In this note we first provide a left duo ring (but not weakly right duo) in spite of it being left Noetherian and local. Thus we observe conditions under which weakly one-sided duo rings may be two-sided. We prove that a weakly one-sided duo ring R is weakly duo under each of the following conditions: (1) R is semilocal with nil Jacobson radical; (2) R is locally finite. Based on the preceding case (1) we study a kind of composition length of a right or left Artinian weakly duo ring R, obtaining that i(R) is finite and $\alpha^{i(R)}R\;=\;R\alpha^{i(R)\;=\;R\alpha^{i(R)}R\;for\;all\;\alpha\;{\in}\;R$, where i(R) is the index (of nilpotency) of R. Note that one-sided Artinian rings and locally finite rings are strongly $\pi-regular$. Thus we also observe connections between strongly $\pi-regular$ weakly right duo rings and related rings, constructing available examples.

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References

  1. G. Azumaya, Strongly $\pi$-regular rings, J. Fac. Sci. Hokkaido Univ. 13 (1954), 34-39
  2. A. Badawi, On abelian $\pi$-regular rings, Comm. Algebra 25 (1997), 1009-1021 https://doi.org/10.1080/00927879708825906
  3. H. Bass, Finitistic homological dimension and a homological generalization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488
  4. G. F. Birkenmeier, J. Y. Kim, and J. K. Park, A connection between weak regularity and the simplicity of prime factor rings, Proc. Amer. Math. Soc. 122 (1994), 53-58 https://doi.org/10.2307/2160840
  5. A. W. Chatters and W. Xue, On right duo p.p. rings, Glasg. Math. J. 32 (1990), 221-225 https://doi.org/10.1017/S0017089500009253
  6. R. C. Courter, Finite dimensional right duo algebras are duo, Proc. Amer. Math. Soc. 84 (1982), 157-161 https://doi.org/10.2307/2043655
  7. F. Dischinger, Sur les anneaux fortement $\pi$-reguliers, C. R. Math. Acad. Sci. Paris 283 (1976), 571-573
  8. K. R. Goodearl, Von Neumann Regular Rings, Pitman, London, 1979
  9. K. R. Goodearl and R. B. Warfield. JR., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, 1989
  10. C. Y. Hong, N. K. Kim, T. K. Kwak, and Y. Lee, On weak $\pi$-regularity of rings whose prime ideals are maximal, J. Pure Appl. Algebra 146 (2000), 35-44 https://doi.org/10.1016/S0022-4049(98)00177-7
  11. C. Huh, H. K. Kim, and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167 (2002), 37-52 https://doi.org/10.1016/S0022-4049(01)00149-9
  12. C. Huh, S. H. Jang, C. O. Kim, and Y. Lee, Rings whose maximal one-sided ideals are two-sided, Bull. Korean Math. Soc. 39 (2002), 411-422
  13. C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), 751-761 https://doi.org/10.1081/AGB-120013179
  14. N. Jacobson, The Theory of Rings, American Mathematical Society, 1943
  15. J. Lambek, Lectures on Rings and Modules, Blaisdell, Waltham, 1966
  16. Y. Lee and C. Huh, On rings in which every maximal one-sided ideal contains a maximal ideal, Comm. Algebra 27 (1999), 3969-3978 https://doi.org/10.1080/00927879908826676
  17. J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, John Wiley & Sons Ltd., 1987
  18. L. H. Rowen, Ring Theory, Academic Press, Inc., 1991
  19. R. M. Thrall, Some generalizations of quasi-Frobenius algebras, Trans. Amer. Math. Soc. 36 (1948), 173-183
  20. H. Tominaga, Some remarks on$\pi$-regular rings of bounded index, Math. J. Okayama Univ. 4 (1955), 135-141 https://doi.org/10.1512/iumj.1955.4.54003
  21. X. Yao, Weakly right duo rings, Pure Appl. Math. Sci. 21 (1985), 19-24
  22. H. -P. Yu, On quasi-duo rings, Glasg. Math. J. 37 (1995), 21-31 https://doi.org/10.1017/S0017089500030342