Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A GENERALIZATION OF SYMMETRIC RING PROPERTY

  • Kim, Hong Kee (Department of Mathematics and RINS Gyeongsang National University) ;
  • Kwak, Tai Keun (Department of Mathematics Daejin University) ;
  • Lee, Seung Ick (Department of Mathematics Pusan National University) ;
  • Lee, Yang (Department of Mathematics Pusan National 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 Pusan National University)
  • Received : 2015.07.22
  • Published : 2016.09.30
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Abstract

This note focuses on a ring property in which upper and lower nilradicals coincide, as a generalizations of symmetric rings. The concept of symmetric ideal and ring in the noncommutative ring theory was initially introduced by Lambek, as an extension of the usual commutative ideal theory. The investigation of symmetric rings provided many useful results to the study in the noncommutative ring theory. So the results obtained from this study may be applicable to observing the structure of zero divisors in various kinds of algebraic systems containing matrix rings and polynomial rings.

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References

  1. D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2272. https://doi.org/10.1080/00927879808826274
  2. 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
  3. 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
  4. 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
  5. 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
  6. U. S. Chakraborty and K. Das, On nil-symmetric rings, J. Math. 2014 (2014), Article ID 483784, 7 pages.
  7. J. L. Dorroh, Concerning adjunctins to algebras, Bull. Amer. Math. Soc. 38 (1932), 85-88. https://doi.org/10.1090/S0002-9904-1932-05333-2
  8. K. E. Eldridge, Orders for finite noncommutative rings with unity, Amer. Math. Monthly 75 (1968), 512-514. https://doi.org/10.2307/2314716
  9. K. R. Goodearl, Von Neumann Regular Rings, Pitman, London, 1979.
  10. K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, Cambridge-New York-Port Chester-Melbourne-Sydney, 1989.
  11. C. Huh, H. K. Kim, N. K. Kim, and Y. Lee, Basic examples and extensions of symmetric rings, J. Pure Appl. Algebra 202 (2005), no. 1-3, 154-167. https://doi.org/10.1016/j.jpaa.2005.01.009
  12. 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
  13. C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative ring, Comm. Algebra 30 (2002), no. 2, 751-761. https://doi.org/10.1081/AGB-120013179
  14. 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
  15. J. Lambek, Lectures on Rings and Modules, Blaisdell Publishing Company, Waltham, 1966.
  16. J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359-368. https://doi.org/10.4153/CMB-1971-065-1
  17. J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, John Wiley & Sons Ltd., New York, 1987.
  18. R. Mohammadi, A. Moussavi, and M. Zahiri, On nil-semicommutative rings, Int. Electron. J. Algebra 11 (2012), 20-37.
  19. L. Motais de Narbonne, Anneaux semi-commutatifs et unis riels anneaux dont les id aux principaux sont idempotents, Proceedings of the 106th National Congress of Learned Societies (Perpignan, 1981), 71-73, Bib. Nat., Paris, 1982.
  20. M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17. https://doi.org/10.3792/pjaa.73.14
  21. J. C. Shepherdson, Inverses and zero-divisors in matrix ring, Proc. London Math. Soc. 3 (1951), 71-85.
  22. G. Y. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43-60. https://doi.org/10.1090/S0002-9947-1973-0338058-9

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