Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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ON A RING PROPERTY RELATED TO NILRADICALS

  • Jin, Hai-lan (Department of Mathematics Yanbian University) ;
  • Piao, Zhelin (Department of Mathematics Yanbian University) ;
  • Yun, Sang Jo (Department of Mathematics Dong-A University)
  • Received : 2018.12.19
  • Accepted : 2019.02.13
  • Published : 2019.03.30
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Abstract

In this article we investigate the structure of rings in which lower nilradicals coincide with upper nilradicals. Such rings shall be said to be quasi-2-primal. It is shown first that the $K{\ddot{o}}the^{\prime}s$ conjecture holds for quasi-2-primal rings. So the results in this article may provide interesting and useful information to the study of nilradicals in various situations. In the procedure we study the structure of quasi-2-primal rings, and observe various kinds of quasi-2-primal rings which do roles in ring theory.

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References

  1. D.D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), 2265-2272. https://doi.org/10.1080/00927879808826274
  2. R. Antoine, Nilpotent elements and Armendariz rings, J. Algebra 319 (2008), 3128-3140. https://doi.org/10.1016/j.jalgebra.2008.01.019
  3. E.P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc. 18 (1974), 470-473. https://doi.org/10.1017/S1446788700029190
  4. G.F. Birkenmeier, H.E. Heatherly, and E.K. Lee, Completely prime ideals and associated radicals, Proc. Biennial Ohio State-Denison Conference 1992, edited by S. K. Jain and S. T. Rizvi, World Scientific, Singapore-New Jersey-London-Hong Kong (1993), 102-129.
  5. G.F. Birkenmeier, J.Y. Kim, and J.K. Park, Regularity conditions and the simplicity of prime factor rings, J. Pure Appl. Algebra 115 (1997), 213-230. https://doi.org/10.1016/S0022-4049(96)00011-4
  6. V. Camillo and P.P. Nielsen, McCoy rings and zero-divisors, J. Pure Appl. Algebra 212 (2008), 599-615. https://doi.org/10.1016/j.jpaa.2007.06.010
  7. V. Camillo, C.Y. Hong, N.K. Kim, Y. Lee, and P.P. Nielsen, Nilpotent ideals in polynomial and power series rings, Proc. Amer. Math. Soc. 138 (2010), 1607-1619. https://doi.org/10.1090/S0002-9939-10-10252-4
  8. K.R. Goodearl, Von Neumann Regular Rings, Pitman, London, 1979.
  9. S.U. Hwang, Y.C, Jeon, and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302 (2006), 186-199. https://doi.org/10.1016/j.jalgebra.2006.02.032
  10. Y.C. Jeon, H.K. Kim, Y. Lee, and J.S. Yoon, On weak Armendariz rings, Bull. Korean Math. Soc. 46 (2009), 135-146. https://doi.org/10.4134/BKMS.2009.46.1.135
  11. N.K. Kim, K.H. Kim, and Y. Lee, Power series rings satisfying a zero divisor property, Comm. Algebra 34 (2006), 2205-2218. https://doi.org/10.1080/00927870600549782
  12. N.K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), 477-488. https://doi.org/10.1006/jabr.1999.8017
  13. G. Kothe, Die Struktur der Ringe deren Restklassenring nach dem Radikal vollstanding irreduzibel ist, Math. Z. 32 (1930), 161-186. https://doi.org/10.1007/BF01194626
  14. J. Krempa, Logical connections among some open problems concerning nil rings, Fund. Math. 76 (1972), 121-130. https://doi.org/10.4064/fm-76-2-121-130
  15. T.Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991.
  16. G. Marks, On 2-primal Ore extensions, Comm. Algebra 29 (2001), 2113-2123. https://doi.org/10.1081/AGB-100002173
  17. G. Marks, A taxonomy of 2-primal rings, J. Algebra 266 (2003), 494-520. https://doi.org/10.1016/S0021-8693(03)00301-6
  18. J.V. Neumann, On regular rings, Proceedngs of the National Academy of Sciences, 22 (1936), 707-713.
  19. M.B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), 14-17.
  20. L.H. Rowen, Ring Theory, Academic Press, San Diego, 1991.
  21. A.D. Sands, Radical and Morita contex, J. Algebra 24 (1973), 335-345. https://doi.org/10.1016/0021-8693(73)90143-9