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GRADED INTEGRAL DOMAINS AND PRÜFER-LIKE DOMAINS

  • Chang, Gyu Whan (Department of Mathematics Education Incheon National University)
  • Received : 2016.09.23
  • Accepted : 2017.03.30
  • Published : 2017.11.01

Abstract

Let $R={\oplus}_{{\alpha}{\in}{\Gamma}}R_{\alpha}$ be an integral domain graded by an arbitrary torsionless grading monoid ${\Gamma}$, ${\bar{R}}$ be the integral closure of R, H be the set of nonzero homogeneous elements of R, C(f) be the fractional ideal of R generated by the homogeneous components of $f{\in}R_H$, and $N(H)=\{f{\in}R{\mid}C(f)_v=R\}$. Let $R_H$ be a UFD. We say that a nonzero prime ideal Q of R is an upper to zero in R if $Q=fR_H{\cap}R$ for some $f{\in}R$ and that R is a graded UMT-domain if each upper to zero in R is a maximal t-ideal. In this paper, we study several ring-theoretic properties of graded UMT-domains. Among other things, we prove that if R has a unit of nonzero degree, then R is a graded UMT-domain if and only if every prime ideal of $R_{N(H)}$ is extended from a homogeneous ideal of R, if and only if ${\bar{R}}_{H{\backslash}Q}$ is a graded-$Pr{\ddot{u}}fer$ domain for all homogeneous maximal t-ideals Q of R, if and only if ${\bar{R}}_{N(H)}$ is a $Pr{\ddot{u}}fer$ domain, if and only if R is a UMT-domain.

References

  1. D. D. Anderson and D. F. Anderson, Divisorial ideals and invertible ideals in a graded integral domain, J. Algebra 76 (1982), no. 2, 549-569. https://doi.org/10.1016/0021-8693(82)90232-0
  2. D. D. Anderson, D. F. Anderson, and M. Zafrullah, Splitting the t-class group, J. Pure Appl. Algebra 74 (1991), no. 1, 17-37. https://doi.org/10.1016/0022-4049(91)90046-5
  3. D. D. Anderson, D. F. Anderson, and M. Zafrullah, The ring D + XDS[X] and t-splitting sets, Arab. J. Sci. Eng. Sect. C Theme Issues 26 (2001), no. 1, 3-16.
  4. D. D. Anderson, E. Houston, and M. Zafrullah, t-linked extensions, the t-class group, and Nagata's theorem, J. Pure Appl. Algebra 86 (1993), no. 2, 109-124. https://doi.org/10.1016/0022-4049(93)90097-D
  5. D. D. Anderson and L. Mahaney, On primary factorizations, J. Pure Appl. Algebra 54 (1988), no. 2-3, 141-154. https://doi.org/10.1016/0022-4049(88)90026-6
  6. D. D. Anderson and M. Zafrullah, Weakly factorial domains and groups of divisibility, Proc. Amer. Math. Soc. 109 (1990), no. 4, 907-913. https://doi.org/10.1090/S0002-9939-1990-1021893-7
  7. D. F. Anderson, Graded Krull domains, Comm. Algebra 7 (1979), no. 1, 79-106. https://doi.org/10.1080/00927877908822334
  8. D. F. Anderson and G. W. Chang, Homogeneous splitting sets of a graded integral domain, J. Algebra 288 (2005), no. 2, 527-544. https://doi.org/10.1016/j.jalgebra.2005.03.007
  9. 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
  10. D. F. Anderson, G. W. Chang, and M. Zafrullah, Graded-Prufer domains, Comm. Algebra, to appear.
  11. E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form D + M, Michigan Math. J. 20 (1973), 79-95. https://doi.org/10.1307/mmj/1029001014
  12. J. Brewer and E. Rutter, D +M constructions with general overrings, Michigan Math. J. 23 (1976), no. 1, 33-42. https://doi.org/10.1307/mmj/1029001619
  13. G. W. Chang, Locally pseudo-valuation domains of the form $D[X]_{N_v}$ , J. Korean Math. Soc. 45 (2008), no. 5, 1405-1416. https://doi.org/10.4134/JKMS.2008.45.5.1405
  14. G. W. Chang, The A+XB[X] construction from Prufer v-multiplication domains, J. Algebra 439 (2015), 417-437. https://doi.org/10.1016/j.jalgebra.2015.05.030
  15. G. W. Chang, Graded integral domains and Nagata rings II, Korean J. Math. 25 (2017), 215-227.
  16. G. W. Chang, T. Dumitrescu, and M. Zafrullah, t-splitting sets in integral domains, J. Pure Appl. Algebra 187 (2004), no. 1-3, 71-86. https://doi.org/10.1016/j.jpaa.2003.07.001
  17. G. W. Chang and M. Fontana, Uppers to zero in polynomial rings and Prufer-like domains, Comm. Algebra 37 (2009), no. 1, 164-192. https://doi.org/10.1080/00927870802243564
  18. G. W. Chang, B. G. Kang, and J. W. Lim, Prufer v-multiplication domains and related domains of the form D + DS[${\Gamma}^*$], J. Algebra 323 (2010), no. 11, 3124-3133. https://doi.org/10.1016/j.jalgebra.2010.03.010
  19. G. W. Chang, H. Kim, and D. Y. Oh, Kaplansky-type theorems in graded integral domains, Bull. Korean Math. Soc. 52 (2015), no. 4, 1253-1268. https://doi.org/10.4134/BKMS.2015.52.4.1253
  20. G. W. Chang and D. Y. Oh, On t-almost Dedekind graded domains, Bull. Korean Math. Soc., to appear.
  21. D. Costa, J. Mott, and M. Zafrullah, The construction $D+XD_S[X]$, J. Algebra 53 (1978), no. 2, 423-439. https://doi.org/10.1016/0021-8693(78)90289-2
  22. D. Dobbs, E. Houston, T. Lucas, M. Roitman, and M. Zafrullah, On t-linked overrings, Comm. Algebra 20 (1992), no. 5, 1463-1488. https://doi.org/10.1080/00927879208824414
  23. D. Dobbs, E. Houston, T. Lucas, and M. Zafrullah, t-linked overrings and Prufer v- multiplication domains, Comm. Algebra 17 (1989), no. 11, 2835-2852. https://doi.org/10.1080/00927878908823879
  24. D. Dobbs, E. Houston, T. Lucas, and M. Zafrullah, t-linked overrings as intersections of localizations, Proc. Amer. Math. Soc. 109 (1990), no. 3, 637-646. https://doi.org/10.1090/S0002-9939-1990-1017000-7
  25. M. Fontana, J. Huckaba, and I. Papick, Prufer Domains, Marcel Dekker, New York, 1997.
  26. M. Fontana, S. Gabelli, and E. Houston, UMT-domains and domains with Prufer integral closure, Comm. Algebra 25 (1998), no. 4, 1017-1039.
  27. S. Gabelli, E. Houston, and T. Lucas, The t#-property for integral domains, J. Pure Appl. Algebra 194 (2004), no. 3, 281-298. https://doi.org/10.1016/j.jpaa.2004.05.002
  28. R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.
  29. R. Gilmer, J. Mott, and M. Zafrullah, t-invertibility and comparability, in: Commutative Ring Theory. Lecture Notes Pure and Appl. Math., pp. 141-150, vol. 153, Marcel Dekker, New York, 1994.
  30. M. Grin, Some results on v-multiplication rings, Canad. J. Math. 19 (1967), 710-722. https://doi.org/10.4153/CJM-1967-065-8
  31. J. Hedstrom and E. Houston, Pseudo-valuation domains, Pacic J. Math. 75 (1978), no. 1, 137-147. https://doi.org/10.2140/pjm.1978.75.137
  32. J. Hedstrom and E. Houston, Some remarks on star-operations, J. Pure Appl. Algebra 18 (1980), no. 1, 37-44. https://doi.org/10.1016/0022-4049(80)90114-0
  33. E. Houston, Uppers to zero in polynomial rings, in: J. W. Brewer, S. Glaz, W. J. Heinzer, B. M. Olberding (Eds.), Multiplicative Ideal Theory in Commutative Algebra, pp. 243-261, A Tribute to the Work of Robert Gilmer, Springer, 2006.
  34. E. Houston and M. Zafrullah, On t-invertibility II, Comm. Algebra 17 (1989), no. 8, 1955-1969. https://doi.org/10.1080/00927878908823829
  35. J. L. Johnson, Integral closure and generalized transforms in graded domains, Pacic J. Math. 107 (1983), no. 1, 173-178. https://doi.org/10.2140/pjm.1983.107.173
  36. B. G. Kang, Prufer v-multiplication domains and the ring $R[X]_{N_v}$ , J. Algebra 123 (1989), no. 1, 151-170. https://doi.org/10.1016/0021-8693(89)90040-9
  37. I. Kaplansky, Commutative Rings, revised ed., Univ. of Chicago, Chicago, 1974.
  38. C. Nastasescu and F. Van Oystaeyen, Graded Ring Theory, Library of Math. vol. 28, North Holland, Amsterdam, 1982.
  39. D. G. Northcott, Lessons on Rings, Modules, and Multiplicities, Cambridge Univ. Press, Cambridge, 1968.
  40. J. Querre, Intersections d'anneaux integres, J. Algebra 43 (1976), 55-60. https://doi.org/10.1016/0021-8693(76)90144-7
  41. P. Sahandi, On quasi-Prufer and UMt domains, Comm. Algebra 42 (2014), 299-305. https://doi.org/10.1080/00927872.2012.714022
  42. P. Sahandi, Characterizations of graded Prufer *-multiplication domains, Korean J. Math. 22 (2014), 181-206. https://doi.org/10.11568/kjm.2014.22.1.181
  43. A. Seidenberg, A note on the dimension theory of rings, Pacic J. Math. 3 (1953), 505-512. https://doi.org/10.2140/pjm.1953.3.505
  44. F. Wang, On induced operations and UMT-domains, Sichuan Shifan Daxue Xuebao Ziran Kexue Ban 27 (2004), no. 1, 1-9.