Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A HALF-CENTERED STAR-OPERATION ON AN INTEGRAL DOMAIN

  • Qiao, Lei (School of Mathematics Sichuan Normal University) ;
  • Wang, Fanggui (School of Mathematics Sichuan Normal University)
  • Received : 2015.09.28
  • Published : 2017.01.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we study the natural star-operation defined by the set of associated primes of principal ideals of an integral domain, which is called the g-operation. We are mainly concerned with the ideal-theoretic properties of this star-operation. In particular, we investigate DG-domains (i.e., integral domains in which each ideal is a g-ideal), which form a proper subclass of the DW-domains. In order to provide some original examples, we examine the transfer of the DG-property to pullbacks. As an application of the g-operation, it is shown that w-divisorial Mori domains can be seen as a Gorenstein analogue of Krull domains.

Keywords

Acknowledgement

Supported by : NSFC, Doctorial Scientic Research Foundation, Sichuan Normal University, Sichuan Provincial Education Department

References

  1. D. D. Anderson, Star-operations induced by overrings, Comm. Algebra 16 (1988), no. 12, 2535-2553. https://doi.org/10.1080/00927879808823702
  2. D. D. Anderson and S. Cook, Two star-operations and their induced lattices, Comm. Algebra 28 (2000), no. 5, 2461-2475. https://doi.org/10.1080/00927870008826970
  3. S. El Baghdadi and S. Gabelli, w-divisorial domains, J. Algebra 285 (2005), no. 1, 335-355. https://doi.org/10.1016/j.jalgebra.2004.11.016
  4. D. Bennis, A note on Gorenstein global dimension of pullback rings, Int. Electron. J. Algebra 8 (2010), 30-44.
  5. J. W. Brewer and W. J. Heinzer, Associated primes of principal ideals, Duke Math. J. 41 (1974), 1-7. https://doi.org/10.1215/S0012-7094-74-04101-5
  6. P.-J. Cahen, Torsion theory and associated primes, Proc. Amer. Math. Soc. 38 (1973), 471-476. https://doi.org/10.1090/S0002-9939-1973-0357384-6
  7. G. W. Chang, Strong Mori domains and the ring $D[X]_{N_v}$, J. Pure Appl. Algebra 197 (2005), no. 1-3, 293-304. https://doi.org/10.1016/j.jpaa.2004.08.036
  8. D. E. Dobbs, E. G. Houston, T. G. Lucas, M. Roitman, and M. Zafrullah, On t-linked overrings, Comm. Algebra 20 (1992), no. 5, 1463-1488. https://doi.org/10.1080/00927879208824414
  9. D. E. Dobbs, E. G. Houston, T. G. 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
  10. D. E. Dobbs, E. G. Houston, T. G. 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
  11. R. M. Fossum, The Divisor Class Group of a Krull Domain, Springer-Verlag, New York-Heidelberg, 1973.
  12. S. Gabelli, Generalized Dedekind domains, Multiplicative Ideal Theory in Commutative Algebra, pp. 189-206, Springer, New York, 2006.
  13. S. Gabelli and E. Houston, Ideal theory in pullbacks, Non-Noetherian commutative ring theory, 199-227, Math. Appl., vol. 520, Kluwer Acad. Publ., Dordrecht, 2000.
  14. S. Gabelli, E. Houston, and G. Picozza, w-divisoriality in polynomial rings, Comm. Algebra 37 (2009), no. 3, 1117-1127. https://doi.org/10.1080/00927870802278982
  15. J. M. Garcia, P. Jara, and E. Santos, Prufer *-multiplication domains and torsion theories, Comm. Algebra 27 (1999), no. 3, 1275-1295. https://doi.org/10.1080/00927879908826493
  16. R. Gilmer, A class of domains in which primary ideals are valuation ideals, Math. Ann. 161 (1965), 247-254. https://doi.org/10.1007/BF01359908
  17. R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.
  18. R. Gilmer and W. J. Heinzer, Intersections of quotient rings of an integral domain, J. Math. Kyoto Univ. 7 (1967), 133-150. https://doi.org/10.1215/kjm/1250524273
  19. S. Glaz and W. V. Vasconcelos, Flat ideals II, Manuscripta Math. 22 (1977), no. 4, 325-341. https://doi.org/10.1007/BF01168220
  20. J. S. Golan, Torsion Theories, Longman Scientific & Technical, Harlow, 1986.
  21. J. R. Hedstrom and E. G. 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
  22. W. Heinzer, Integral domains in which each non-zero ideal is divisorial, Mathematika 15 (1968), 164-170. https://doi.org/10.1112/S0025579300002527
  23. E. Houston and M. Zafrullah, Integral domains in which each t-ideal is divisorial, Michigan Math. J. 35 (1988), no. 2, 291-300. https://doi.org/10.1307/mmj/1029003756
  24. E. Houston and M. Zafrullah, On t-invertibility. II, Comm. Algebra 17 (1989), no. 8, 1955-1969. https://doi.org/10.1080/00927878908823829
  25. K. Hu and F. Wang, Some results on Gorenstein Dedekind domains and their factor rings, Comm. Algebra 41 (2013), no. 1, 284-293. https://doi.org/10.1080/00927872.2011.629268
  26. J. A. Huckaba and I. J. Papick, A localization of R[x], Canad. J. Math. 33 (1981), no. 1, 103-115. https://doi.org/10.4153/CJM-1981-010-6
  27. 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
  28. I. Kaplansky, Commutative Rings (Revised edition), Univ. Chicago Press, Chicago, 1974.
  29. N.Mahdou and M. Tamekkante, On (strongly) Gorenstein (semi) hereditary rings, Arab. J. Sci. Eng. 36 (2011), no. 3, 431-440. https://doi.org/10.1007/s13369-011-0047-7
  30. A. Mimouni, TW-domains and strong Mori domains, J. Pure Appl. Algebra 177 (2003), no. 1, 79-93. https://doi.org/10.1016/S0022-4049(02)00171-8
  31. A. Mimouni, Integral domains in which each ideal is a W-ideal, Comm. Algebra 33 (2005), no. 5, 1345-1355. https://doi.org/10.1081/AGB-200058369
  32. J. L. Mott and M. Zafrullah, On Prufer v-multiplication domains, Manuscripta Math. 35 (1981), no. 1-2, 1-26. https://doi.org/10.1007/BF01168446
  33. I. J. Papick, Super-primitive elements, Pacific J. Math. 105 (1983), no. 1, 217-226. https://doi.org/10.2140/pjm.1983.105.217
  34. M. H. Park and F. Tartarone, Divisibility properties related to star-operations on integral domains, Int. Electron. J. Algebra 12 (2012), 53-74.
  35. G. Picozza and F. Tartarone, When the semistar operation ${\-}{\star}$ is the identity, Comm. Algebra 36 (2008), no. 5, 1954-1975. https://doi.org/10.1080/00927870801941895
  36. L. Qiao and F. Wang, A Gorenstein analogue of a result of Bertin, J. Algebra Appl. 14 (2015), no. 2, 1550019, 13 pp. https://doi.org/10.1142/S021949881550019X
  37. L. Qiao and F. Wang, A hereditary torsion theory for modules over integral domains and its applications, Comm. Algebra 44 (2016), no. 4, 1574-1587. https://doi.org/10.1080/00927872.2015.1027367
  38. B. Stenstrom, Rings of Quotients, Springer-Verlag, Berlin, 1975.
  39. H. Tang, Gauss' Lemma, Proc. Amer. Math. Soc. 35 (1972), 372-376.
  40. W. V. Vasconcelos, Reflexive modules over Gorenstein rings, Proc. Amer. Math. Soc. 19 (1968), 1349-1355. https://doi.org/10.1090/S0002-9939-1968-0237480-2
  41. W. V. Vasconcelos, Quasi-normal rings, Illinois J. Math. 14 (1970), 268-273.
  42. F. Wang, w-dimension of domains, Comm. Algebra 27 (1999), no. 5, 2267-2276. https://doi.org/10.1080/00927879908826564
  43. F. Wang, w-dimension of domains. II, Comm. Algebra 29 (2001), no. 6, 2419-2428. https://doi.org/10.1081/AGB-100002398
  44. F. Wang and H. Kim, Two generalizations of projective modules and their applications, J. Pure Appl. Algebra 219 (2015), no. 6, 2099-2123. https://doi.org/10.1016/j.jpaa.2014.07.025
  45. F. 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
  46. F. 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
  47. F. Wang and L. Qiao, The w-weak global dimension of commutative rings, Bull. Korean Math. Soc. 52 (2015), no. 4, 1327-1338. https://doi.org/10.4134/BKMS.2015.52.4.1327
  48. M. Zafrullah, The v-operation and intersections of quotient rings of integral domains, Comm. Algebra 13 (1985), no. 8, 1699-1712. https://doi.org/10.1080/00927878508823247
  49. M. Zafrullah, Ascending chain conditions and star operations, Comm. Algebra 17 (1989), no. 6, 1523-1533. https://doi.org/10.1080/00927878908823804