International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2006 / Article

Open Access

Volume 2006 |Article ID 090868 | https://doi.org/10.1155/IJMMS/2006/90868

Najib Mahdou, "On n-flat modules and n-Von Neumann regular rings", International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 090868, 6 pages, 2006. https://doi.org/10.1155/IJMMS/2006/90868

On n-flat modules and n-Von Neumann regular rings

Received04 May 2006
Revised20 Jun 2006
Accepted21 Aug 2006
Published29 Oct 2006

Abstract

We show that each R-module is n-flat (resp., weakly n-flat) if and only if R is an (n,n1)-ring (resp., a weakly (n,n1)-ring). We also give a new characterization of n-Von Neumann regular rings and a characterization of weak n-Von Neumann regular rings for (CH)-rings and for local rings. Finally, we show that in a class of principal rings and a class of local Gaussian rings, a weak n-Von Neumann regular ring is a (CH)-ring.

References

  1. J. Chen and N. Ding, “On n-coherent rings,” Communications in Algebra, vol. 24, no. 10, pp. 3211–3216, 1996. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  2. D. L. Costa, “Parameterizing families of non-Noetherian rings,” Communications in Algebra, vol. 22, no. 10, pp. 3997–4011, 1994. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  3. D. L. Costa and S.-E. Kabbaj, “Classes of D+M rings defined by homological conditions,” Communications in Algebra, vol. 24, no. 3, pp. 891–906, 1996. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  4. D. E. Dobbs, S.-E. Kabbaj, and N. Mahdou, “n-coherent rings and modules,” in Commutative Ring Theory (Fès, 1995), vol. 185 of Lecture Notes in Pure and Appl. Math., pp. 269–281, Marcel Dekker, New York, 1997. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  5. D. E. Dobbs, S.-E. Kabbaj, N. Mahdou, and M. Sobrani, “When is D+Mn-coherent and an (n,d)-domain?,” in Advances in Commutative Ring Theory (Fez, 1997), vol. 205 of Lecture Notes in Pure and Appl. Math., pp. 257–270, Marcel Dekker, New York, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  6. S. Glaz, “The weak dimensions of Gaussian rings,” Proceedings of the American Mathematical Society, vol. 133, no. 9, pp. 2507–2513, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. J. A. Huckaba, Commutative Rings with Zero Divisors, vol. 117 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 1988. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  8. S.-E. Kabbaj and N. Mahdou, “Trivial extensions of local rings and a conjecture of Costa,” in Commutative Ring Theory and Applications (Fez, 2001), vol. 231 of Lecture Notes in Pure and Appl. Math., pp. 301–311, Marcel Dekker, New York, 2003. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  9. S.-E. Kabbaj and N. Mahdou, “Trivial extensions defined by coherent-like conditions,” Communications in Algebra, vol. 32, no. 10, pp. 3937–3953, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. N. Mahdou, “On Costa's conjecture,” Communications in Algebra, vol. 29, no. 7, pp. 2775–2785, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. N. Mahdou, “Steinitz properties in trivial extensions of commutative rings,” The Arabian Journal for Science and Engineering, vol. 26, no. 1, pp. 119–125, 2001. View at: Google Scholar | MathSciNet

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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