Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON φ-VON NEUMANN REGULAR RINGS

  • Zhao, Wei (College of Mathematics Sichuan Normal University) ;
  • Wang, Fanggui (College of Mathematics Sichuan Normal University) ;
  • Tang, Gaohua (School of Mathematical Sciences Guangxi Teachers Education University)
  • Received : 2012.04.07
  • Published : 2013.01.01
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Abstract

Let R be a commutative ring with $1{\neq}0$ and let $\mathcal{H}$ = {R|R is a commutative ring and Nil(R) is a divided prime ideal}. If $R{\in}\mathcal{H}$, then R is called a ${\phi}$-ring. In this paper, we introduce the concepts of ${\phi}$-torsion modules, ${\phi}$-flat modules, and ${\phi}$-von Neumann regular rings.

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References

  1. D. F. Anderson and A. Badawi, On ${\phi}$-Prufer rings and -Bezout rings, Houston J. Math. 30 (2004), no. 2, 331-343.
  2. D. F. Anderson and A. Badawi, on ${\phi}$-Dedekind rings and ${\phi}$-Krull rings, Houston J. Math. 31 (2005), no. 4, 1007-1022.
  3. A. Badawi, On divided commutative rings, Comm. Algebra 27 (1999), no. 3, 1465-1474. https://doi.org/10.1080/00927879908826507
  4. A. Badawi, On ${\phi}$-pseudo-valuation rings, Advances in commutative ring theory (Fez, 1997), 101-110, Lecture Notes in Pure and Appl. Math., 205, Dekker, New York, 1999.
  5. A. Badawi, On ${\phi}$-pseudo-valuation rings. II, Houston J. Math. 26 (2000), no. 3, 473-480.
  6. A. Badawi, On ${\phi}$-chained rings and -pseudo-valuation rings, Houston J. Math. 27 (2001), no. 4, 725-736.
  7. A. Badawi, On divided rings and ${\phi}$-pseudo-valuation rings, Commutative rings, 5-14, Nova Sci. Publ., Hauppauge, NY, 2002.
  8. A. Badawi, On Nonnil-Noetherian rings, Comm. Algebra 31 (2003), no. 4, 1669-1677. https://doi.org/10.1081/AGB-120018502
  9. A. Badawi, Factoring nonnil ideals as a product of prime and invertible ideals, Bulletin of the London Matth. Society 37 (2005), 665-672. https://doi.org/10.1112/S0024609305004509
  10. A. Badawi, On rings with divided nil ideal: a survey, Commutative algebra and its applications, 21-40, Walter de Gruyter, Berlin, 2009.
  11. A. Badawi and D. E. Dobbs, Strong ring extensions and ${\phi}$-pseudo-valuation rings, Houston J. Math. 32 (2006), no. 2, 379-398.
  12. A. Badawi and A. Jaballah, Some finiteness conditions on the set of overrings of a ${\phi}$-ring, Houston J. Math. 34 (2008), no. 2, 397-408.
  13. A. Badawi and T. G. Lucas, Rings with prime nilradical, in Arithmetical Properties of Commutative Rings and Monoids, vol. 241 of Lect. Notes Pure Appl. Math., pp. 198-212, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2005.
  14. A. Badawi and T. G. Lucas, on ${\phi}$-Mori rings, Houston J. Math. 32 (2006), no. 1, 1-32.
  15. D. E. Dobbs, Divided rings and going-down, Pacific J. Math. 67 (1976), no. 2, 353-363. https://doi.org/10.2140/pjm.1976.67.353
  16. S. Hizem and A. Benhissi, Nonnil-Noetherian rings and the SFT property, Rocky Moun-tain J. Math. 41 (2011), no. 5, 1483-1500. https://doi.org/10.1216/RMJ-2011-41-5-1483
  17. J. A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York/Basel, 1988.
  18. H. Kim and F. G. Wang, On ${\phi}$-strong Mori rings, Houston J. Math. 38 (2012), no. 2, 359-371.
  19. C. Lomp and A. Sant'ana, Comparability, distributivity and non-commutative -rings, Groups, rings and group rings, 205-217, Contemp. Math., 499, Amer. Math. Soc., Providence, RI, 2009.
  20. F. G. Wang, Commutative Rings and Star-Operation Theory, Sicence Press, Beijing, 2006.

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