DOI QR코드

DOI QR Code

ON STRONGLY 2-PRIMAL RINGS

Hwang, Seo-Un;Lee, Yang;Park, Kwang-Sug

  • Received : 2007.08.08
  • Published : 2007.12.25

Abstract

We first find strongly 2-primal rings whose sub direct product is not (strongly) 2-primal. Moreover we observe some kinds of ring extensions of (strongly) 2-primal rings. As an example we show that if R is a ring and M is a multiplicative monoid in R consisting of central regular elements, then R is strongly 2-primal if and only if so is $RM^{-1}$. Various properties of (strongly) 2-primal rings are also studied.

Keywords

(strongly) 2-primal ring;prime radical;sub direct product;ring extension

References

  1. 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.
  2. 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
  3. Y.U. Cho, N.K. Kim, M.H. Kwon, and Y. Lee, Classical quotient rings and ordinary extensions of 2-primal rings, Algebra Colloq. 13 (2006), 513-523. https://doi.org/10.1142/S1005386706000460
  4. K.R. Goodearl, Von Neumann Regular Rings, Pitman, London, 1979.
  5. K.Y. Ham, C. Huh, Y.C. Hwang, Y.C. Jeon, H.K. Kim, S.M. Lee, Y. Lee, S.R. O, and J.S. Yoon, On weak Armendariz rings (Submitted).
  6. Y. Hirano, Some studies on strongly ${\pi}$-regular rings, Math. J. Okayama Univ. 20 (1978), 141-149.
  7. C.Y. Hong, H.K. Kim, N.K. Kim, T.K. Kwak, Y. Lee, and K.S. Park, Rings whose nilpotent elements form a Levitzki radical ring, Comm. Algebra (To appear).
  8. 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
  9. N.K. Kim and Y. Lee, On rings whose prime ideals are completely prime, J. Pure Appl. Algebra 170 (2002), 255-265. https://doi.org/10.1016/S0022-4049(01)00148-7
  10. C. Huh, H.K. Kim and Y. Lee, Questions on 2-primal rings, Comm. Algebra 26(2) (1998), 595-600. https://doi.org/10.1080/00927879808826150
  11. C. Huh, H.K. Kim, 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. G. Marks, Direct product and power series formations over 2-primal rings, Advances in Ring Theory, edited by S.K. Jain and S. Tariq rizvi, Birkhauser, Boston-Basel-Berlin (1997), 239-245.
  13. G. Marks, Skew polynomial rings over 2-primal rings, Comm. Algebra 27(9) (1999), 4411-442. https://doi.org/10.1080/00927879908826705
  14. G. Marks, On 2-primal Ore extensions, Comm. Algebra 29 (2001), 2113-2123. https://doi.org/10.1081/AGB-100002173
  15. G. Marks, Reversible and symmetric rings, J. Pure Appl. Algebra 174 (2002), 311-318. https://doi.org/10.1016/S0022-4049(02)00070-1
  16. J.C. McConnell and J.C. Robson, Noncommutative Noetherian Rings, John Wiley & Sons Ltd., 1987.
  17. L.H. Rowen, Examples of semiperfect rings, Israel J. Math. 65(3) (1989), 273-283. https://doi.org/10.1007/BF02764865
  18. L.H. Rowen, Ring Theory, Academic Press, Inc., 1991.
  19. G. Shin, Prime ideals and sheaf representation of a pseudo symmetric rings, Trans. Amer. Math. Soc. 184 (1973), 43-60. https://doi.org/10.1090/S0002-9947-1973-0338058-9
  20. S.-H. Sun, Noncommutative rings in which every prime ideal is contained m a unique maximal ideal, J. Pure Appl. Algebra 76 (1991), 179-192. https://doi.org/10.1016/0022-4049(91)90060-F
  21. X. Yao, Weakly right duo rings, Pure and Appl. Math. Sci. XXI (1985), 19-24.

Cited by

  1. ARMENDARIZ PROPERTY OVER PRIME RADICALS vol.50, pp.5, 2013, https://doi.org/10.4134/JKMS.2013.50.5.973