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ON WEAK II-REGULARITY AND THE SIMPLICITY OF PRIME FACTOR RINGS

  • Kim, Jin-Yong (Department of Mathematics and Institute of Natural Sciences Kyung Hee University) ;
  • Jin, Hai-Lan (Department of Mathematics Yanbian University)
  • Published : 2007.02.28

Abstract

A connection between weak ${\pi}-regularity$ and the condition every prime ideal is maximal will be investigated. We prove that a certain 2-primal ring R is weakly ${\pi}-regular$ if and only if every prime ideal is maximal. This result extends several known results nontrivially. Moreover a characterization of minimal prime ideals is also considered.

Keywords

completely prime ideals;2-primal rings;weakly ${\pi}-regular$ rings;pseudo symmetric rings;minimal prime ideals

References

  1. 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), no. 1, 53-58 https://doi.org/10.1090/S0002-9939-1994-1231028-7
  2. G. F. Birkenmeier, Regularity conditions and the simplicity of prime factor rings, J. Pure Appl. Algebra 115 (1997), no. 3, 213-230 https://doi.org/10.1016/S0022-4049(96)00011-4
  3. G. F. Birkenmeier, A characterization of minimal prime ideals, Glasgow Math. J. 40 (1998), no. 2, 223-236 https://doi.org/10.1017/S0017089500032547
  4. V. Camillo and Y. Xiao, Weakly regular rings, Comm. Algebra 22 (1994), no. 10, 4095- 4112 https://doi.org/10.1080/00927879408825068
  5. V. R. Chandran, On two analogues of Cohen's theorem, Indian J. Pure Appl. Math. 8 (1977), no. 1, 54-59
  6. J. W. Fisher and R. L. Snider, On the von Neumann regularity of rings with regular prime factor rings, Pacific J. Math. 54 (1974), 135-144 https://doi.org/10.2140/pjm.1974.54.135
  7. V. Gupta, Weakly $\pi$-regular rings and group rings, Math. J. Okayama Univ. 19 (1976/77), no. 2, 123-127
  8. Y. Hirano, Some studies on strongly $\pi$-regular rings, Math. J. Okayama Univ. 20 (1978), no. 2, 141-149
  9. 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
  10. V. S. Ramamurthi, Weakly regular rings, Canad. Math. Bull. 16 (1973), 317-321 https://doi.org/10.4153/CMB-1973-051-7
  11. G. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43-60 https://doi.org/10.2307/1996398
  12. S. H. Sun, Noncommutative rings in which every prime ideal is contained in a unique maximal ideal, J. Pure Appl. Algebra 76 (1991), no. 2, 179-192 https://doi.org/10.1016/0022-4049(91)90060-F
  13. X. Yao, Weakly right duo rings, Pure Appl. Math. Sci. 21 (1985), no. 1-2, 19-24
  14. G. F. Birkenmeier, H. E. Heatherly, and Enoch K. Lee, Completely prime ideals and associated radicals, Ring theory (Granville, OH, 1992), 102-129, World Sci. Publ., River Edge, NJ, 1993
  15. I. S. Cohen, Commutative rings with restricted minimum condition, Duke Math. J. 17 (1950). 27-42 https://doi.org/10.1215/S0012-7094-50-01704-2
  16. H. H. Storrer, Epimorphismen von kommutativen Ringen, Comment. Math. Helv. 43 (1968), 378-401 https://doi.org/10.1007/BF02564404

Cited by

  1. SOME STUDIES ON 2-PRIMAL RINGS, (S,1)-RINGS AND THE CONDITION (KJ) vol.25, pp.3, 2010, https://doi.org/10.4134/CKMS.2010.25.3.343