Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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GRADED INTEGRAL DOMAINS AND NAGATA RINGS, II

  • Chang, Gyu Whan (Department of Mathematics Education Incheon National University)
  • Received : 2017.03.07
  • Accepted : 2017.06.01
  • Published : 2017.06.30
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Abstract

Let D be an integral domain with quotient field K, X be an indeterminate over D, K[X] be the polynomial ring over K, and $R=\{f{\in}K[X]{\mid}f(0){\in}D\}$; so R is a subring of K[X] containing D[X]. For $f=a_0+a_1X+{\cdots}+a_nX^n{\in}R$, let C(f) be the ideal of R generated by $a_0$, $a_1X$, ${\ldots}$, $a_nX^n$ and $N(H)=\{g{\in}R{\mid}C(g)_{\upsilon}=R\}$. In this paper, we study two rings $R_{N(H)}$ and $Kr(R,{\upsilon})=\{{\frac{f}{g}}{\mid}f,g{\in}R,\;g{\neq}0,{\text{ and }}C(f){\subseteq}C(g)_{\upsilon}\}$. We then use these two rings to give some examples which show that the results of [4] are the best generalizations of Nagata rings and Kronecker function rings to graded integral domains.

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References

  1. D.D. Anderson, Some remarks on the ring R(X), Comment. Math. Univ. St. Paul. 26 (1977), 137-140.
  2. D.D. Anderson and D.F. Anderson, Divisorial ideals and invertible ideals in a graded integral domain, J. Algebra 76 (1982), 549-569. https://doi.org/10.1016/0021-8693(82)90232-0
  3. D.F. Anderson and G.W. Chang, Homogeneous splitting sets of a graded integral domain, J. Algebra 288 (2005), 527-544. https://doi.org/10.1016/j.jalgebra.2005.03.007
  4. D.F. Anderson and G.W. Chang, Graded integral domains and Nagata rings, J. Algebra 387 (2013), 169-184. https://doi.org/10.1016/j.jalgebra.2013.04.021
  5. J.T. Arnold, On the ideal theory of the Kronecker function ring and the domain D(X), Canad. J. Math. 21 (1969), 558-563. https://doi.org/10.4153/CJM-1969-063-4
  6. G.W. Chang, Prufer *-multiplication domains, Nagata rings, and Kronecker function rings, J. Algebra 319 (2008), 309-319. https://doi.org/10.1016/j.jalgebra.2007.10.010
  7. L.G. Chouinard II, Krull semigroups and divisor class groups, Canad. J. Math. 33 (1981), 1459-1468. https://doi.org/10.4153/CJM-1981-112-x
  8. D. Costa, J. Mott, and M. Zafrullah, The construction D+XDS[X], J. Algebra 53 (1978), 423-439. https://doi.org/10.1016/0021-8693(78)90289-2
  9. F. Decruenaere and E. Jespers, Prufer domains and graded rings, J. Algebra 53 (1992), 308-320.
  10. M. Fontana and K.A. Loper, A historical overview of Kronecker function rings, Nagata rings, and related star and semistar operations, in: J.W. Brewer, S. Glaz, W.J. Heinzer, B.M. Olberding (Eds.), Multiplicative Ideal Theory in Commu-tative Algebra. A Tribute to the Work of Robert Gilmer, Springer, 2006, pp. 169-187.
  11. R. Gilmer, An embedding theorem for HCF-rings, Proc. Cambridge Philos. Soc. 68 (1970), 583-587. https://doi.org/10.1017/S0305004100076568
  12. R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.
  13. 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
  14. D.G. Northcott, A generalization of a theorem on the content of polynomials, Proc. Cambridge Philos. Soc. 55 (1959), 282-288. https://doi.org/10.1017/S030500410003406X
  15. D.G. Northcott, Lessons on Rings, Modules, and Multiplicities, Cambridge Univ. Press, Cambridge, 1968.
  16. P. Sahandi, Characterizations of graded Prufer *-multiplication domain, Korean J. Math. 22 (2014), 181-206. https://doi.org/10.11568/kjm.2014.22.1.181

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