Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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INTEGRAL CLOSURE OF A GRADED NOETHERIAN DOMAIN

  • Received : 2009.11.12
  • Accepted : 2010.06.10
  • Published : 2011.05.01
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Abstract

We show that, if R is a graded Noetherian ring and I is a proper ideal of R generated by n homogeneous elements, then any prime ideal of R minimal over I has h-height ${\leq}$ n, and that if R is a graded Noetherian domain with h-dim R ${\leq}$ 2, then the integral closure R' of R is also a graded Noetherian domain with h-dim R' ${\leq}$ 2. We also present a short improved proof of the result that, if R is a graded Noetherian domain, then the integral closure of R is a graded Krull domain.

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References

  1. D. D. Anderson and D. F. Anderson, Divisibility properties of graded domains, Canad. J. Math. 34 (1982), no. 1, 196-215. https://doi.org/10.4153/CJM-1982-013-3
  2. A. Bouvier and G. Germain, A concise proof of a theorem of Mori-Nagata, C. R. Math. Rep. Acad. Sci. Canada 1 (1978/79), no. 6, 315-318.
  3. L. Budach, Struktur Noetherscher kommutativer Halbgruppen, Monatsb. Deutsch. Akad. Wiss. Berlin 6 (1964), 85-88.
  4. R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972.
  5. S. Goto and K. Watanabe, On graded rings. I, J. Math. Soc. Japan 30 (1978), no. 2, 179-213. https://doi.org/10.2969/jmsj/03020179
  6. S. Goto and K. Watanabe, On graded rings. II. (Zn-graded rings), Tokyo J. Math. 1 (1978), no. 2, 237-261. https://doi.org/10.3836/tjm/1270216496
  7. S. Goto and K. Yamagishi, Finite generation of Noetherian graded rings, Proc. Amer. Math. Soc. 89 (1983), no. 1, 41-44. https://doi.org/10.1090/S0002-9939-1983-0706507-1
  8. W. Heinzer, On Krull overrings of a Noetherian domain, Proc. Amer. Math. Soc. 22 (1969), 217-222. https://doi.org/10.1090/S0002-9939-1969-0254022-7
  9. W. Heinzer and M. Roitman, The homogeneous spectrum of a graded commutative ring, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1573-1580. https://doi.org/10.1090/S0002-9939-01-06231-1
  10. B. G. Kang, Integral closure of rings with zero divisors, J. Algebra 162 (1993), no. 2, 317-323. https://doi.org/10.1006/jabr.1993.1256
  11. I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
  12. H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, Cambridge, 1986.
  13. M. Nagata, Local Rings, Interscience, New York, 1962.
  14. J. Nishimura, Note on integral closures of a Noetherian integral domain, J. Math. Kyoto Univ. 16 (1976), no. 1, 117-122. https://doi.org/10.1215/kjm/1250522963
  15. J. S. Okon, D. E. Rush, and L. J. Wallace, A Mori-Nagata theorem for lattices and graded rings, Houston J. Math. 31 (2005), no. 4, 973-997.
  16. M. H. Park, Integral closure of graded integral domains, Comm. Algebra 35 (2007), no. 12, 3965-3978. https://doi.org/10.1080/00927870701509511
  17. L. J. Ratliff Jr. and D. E. Rush, Two notes on homogeneous prime ideals in graded Noetherian rings, J. Algebra 264 (2003), no. 1, 211-230. https://doi.org/10.1016/S0021-8693(03)00102-9
  18. D. E. Rush, Noetherian properties in monoid rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 259-278. https://doi.org/10.1016/S0022-4049(03)00103-8
  19. O. Zariski and P. Samuel, Commutative Algebra. Vol. I, Graduate Texts in Mathematics, Vol. 28. Springer-Verlag, New York-Heidelberg-Berlin, 1975.

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