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A Note on S-Noetherian Domains

LIM, JUNG WOOK

  • Received : 2014.09.10
  • Accepted : 2014.11.07
  • Published : 2015.09.23

Abstract

Let D be an integral domain, t be the so-called t-operation on D, and S be a (not necessarily saturated) multiplicative subset of D. In this paper, we study the Nagata ring of S-Noetherian domains and locally S-Noetherian domains. We also investigate the t-Nagata ring of t-locally S-Noetherian domains. In fact, we show that if S is an anti-archimedean subset of D, then D is an S-Noetherian domain (respectively, locally S-Noetherian domain) if and only if the Nagata ring $D[X]_N$ is an S-Noetherian domain (respectively, locally S-Noetherian domain). We also prove that if S is an anti-archimedean subset of D, then D is a t-locally S-Noetherian domain if and only if the polynomial ring D[X] is a t-locally S-Noetherian domain, if and only if the t-Nagata ring $D[X]_{N_v}$ is a t-locally S-Noetherian domain.

Keywords

S-Noetherian domain;(t-)locally S-Noetherian domain;(t-)Nagata ring;finite (t-)character

References

  1. D. D. Anderson, B. G. Kang, and M. H. Park, Anti-archimedean rings and power series rings, Comm. Algebra, 26(1998), 3223-3238. https://doi.org/10.1080/00927879808826338
  2. D. D. Anderson, D. J. Kwak, and M. Zafrullah, Agreeable domains, Comm. Algebra, 23(1995), 4861-4883. https://doi.org/10.1080/00927879508825505
  3. M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company, 1969.
  4. G. W. Chang, Strong Mori domains and the ring $D[X]_{N_v}$, J. Pure Appl. Algebra, 197(2005), 293-304. https://doi.org/10.1016/j.jpaa.2004.08.036
  5. M. Fontana, S. Gabelli, and E. Houston, UMT-domains and domains with Prufer integral closure, Comm. Algebra, 26(1998), 1017-1039. https://doi.org/10.1080/00927879808826181
  6. R. Gilmer, Multiplicative Ideal Theory, Queen's Papers in Pure Appl. Math., 90, Queen's University, Kingston, Ontario, 1992.
  7. B. G. Kang, Prufer v-multiplication domains and the ring $R[X]_{N_v}$, J. Algebra, 123(1989), 151-170. https://doi.org/10.1016/0021-8693(89)90040-9
  8. I. Kaplansky, Commutative Rings, Polygonal Publishing House, Washington, New Jersey, 1994.
  9. H. Kim, M. O. Kim, and J. W. Lim, On S-strong Mori domains, J. Algebra, 416(2014), 314-332. https://doi.org/10.1016/j.jalgebra.2014.06.015
  10. J. W. Lim and D. Y. Oh, S-Noetherian properties of composite ring extensions, Comm. Algebra, 43(2015), 2820-2829. https://doi.org/10.1080/00927872.2014.904329
  11. J. W. Lim and D. Y. Oh, S-Noetherian properties on amalgamated algebras along an ideal, J. Pure Appl. Algebra, 218(2014), 1075-1080. https://doi.org/10.1016/j.jpaa.2013.11.003
  12. Z. Liu, On S-Noetherian rings, Arch. Math. (Brno), 43(2007), 55-60.
  13. F. Wang and R. L. McCasland, On w-modules over strong Mori domains, Comm. Algebra, 25(1997), 1285-1306. https://doi.org/10.1080/00927879708825920
  14. D. D. Anderson, D. F. Anderson, and R. Markanda, The ring R(X) and R(X), J. Algebra, 95(1985), 96-115. https://doi.org/10.1016/0021-8693(85)90096-1
  15. D. D. Anderson and T. Dumitrescu, S-Noetherian rings, Comm. Algebra, 30(2002), 4407-4416. https://doi.org/10.1081/AGB-120013328

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