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A NOTE ON w-NOETHERIAN RINGS

  • Xing, Shiqi (College of Mathematics and Software Science Sichuan Normal University) ;
  • Wang, Fanggui (College of Mathematics and Software Science Sichuan Normal University)
  • Received : 2014.03.06
  • Published : 2015.03.31

Abstract

Let R be a commutative ring. An R-module M is called a w-Noetherian module if every submodule of M is of w-finite type. R is called a w-Noetherian ring if R as an R-module is a w-Noetherian module. In this paper, we present an exact version of the Eakin-Nagata Theorem on w-Noetherian rings. To do this, we prove the Formanek Theorem for w-Noetherian rings. Further, we point out by an example that the condition (${\dag}$) in the Chung-Ha-Kim version of the Eakin-Nagata Theorem on SM domains is essential.

Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. D. D. Anderson and S. J. Cook, Two star operations and their induced lattices, Comm. Algebra 29 (2000), no. 5, 2461-2475.
  2. G. W. Chang and M. Zafrullah, The w-integral closure of integral domains, J. Algebra 295 (2006), no. 1, 195-210. https://doi.org/10.1016/j.jalgebra.2005.04.025
  3. W. Chung, J. Ha, and H. Kim, Some remarks on strong Mori domains, Houston J. Math. 38 (2012), no. 4, 1051-1059.
  4. P. M. Eakin, The converse to a well known theorem on Noetherian rings, Math. Ann. 177 (1968), 278-282. https://doi.org/10.1007/BF01350720
  5. E. Formanek, Die statze von Bertini fur lokale Ringe, Math. Ann. 229 (1977), 97-111. https://doi.org/10.1007/BF01351596
  6. H. Kim, E. S. Kim, and Y. S. Park, Injective modules over strong Mori domains, Houston J. Math. 34 (2008), no. 2, 349-360.
  7. M. Nagata, A type of subrings of a Noetherian ring, J. Math. Kyoto Univ. 8 (1968), 465-467.
  8. M. H. Park, Groups rings and semigroup rings over strong Mori domains, J. Pure Appl. Algebra 163 (2001), no. 3, 301-318. https://doi.org/10.1016/S0022-4049(00)00160-2
  9. M. H. Park, On overrings of Strong Mori domains, J. Pure Appl. Algebra 172 (2002), no. 1, 79-85. https://doi.org/10.1016/S0022-4049(01)00135-9
  10. F. G. Wang, Finitely presented type modules and w-coherent rings, J. Sichuan Normal Univ. 33 (2010), 1-9.
  11. F. G. Wang, and R. L. McCasland, On w-modules over strong Mori domains, Comm. Algebra 25 (1997), no. 4, 1285-1306. https://doi.org/10.1080/00927879708825920
  12. F. G. Wang, and R. L. McCasland, On strong Mori domains, J. Pure Appl. Algebra 135 (1999), no. 2, 155-165. https://doi.org/10.1016/S0022-4049(97)00150-3
  13. F. G. Wang, J. Zhang, Injective modules over w-Noetherian rings, Acta math. Sinica (Chin. Ser.) 53 (2010), no. 6, 1119-1130.
  14. L. Xi, F. G. Wang, and Y. Tian, On w-linked overrings, J. Math. Res. Exposition 31 (2011), 337-346.
  15. H. Y. Yin, F. G. Wang, X. S. Zhu, and Y. H. Chen, w-modules over commutative rings, J. Korean Math. Soc. 48 (2011), no. 1, 146-151.

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