Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

LOCALLY PSEUDO-VALUATION DOMAINS OF THE FORM D[X]Nv

  • Published : 2008.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Let D be an integral domain, X an indeterminate over D, $N_v = \{f{\in}D[X]|(A_f)_v=D\}.$. Among other things, we introduce the concept of t-locally PVDs and prove that $D[X]N_v$ is a locally PVD if and only if D is a t-locally PVD and a UMT-domain, if and only if D[X] is a t-locally PVD, if and only if each overring of $D[X]N_v$ is a locally PVD.

Keywords

References

  1. D. F. Anderson, Comparability of ideals and valuation overrings, Houston J. Math. 5 (1979), no. 4, 451-463
  2. D. F. Anderson and A. Ryckaert, The class group of D+M, J. Pure Appl. Algebra 52 (1988), no. 3, 199-212 https://doi.org/10.1016/0022-4049(88)90091-6
  3. J. 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
  4. A. Badawi, Remarks on pseudo-valuation rings, Comm. Algebra 28 (2000), no. 5, 2343-2358 https://doi.org/10.1080/00927870008826964
  5. V. Barucci, On a class of Mori domains, Comm. Algebra 11 (1983), no. 17, 1989-2001 https://doi.org/10.1080/00927878308822944
  6. 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
  7. P.-J. Cahen, J.-L. Chabert, D. E. Dobbs, and F. Tartarone, On locally divided domains of the form Int(D), Arch. Math. (Basel) 74 (2000), no. 3, 183-191 https://doi.org/10.1007/s000130050429
  8. 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
  9. G. W. Chang and M. Zafrullah, The w-integral closure of integral domains, J. Algebra 295 (2006), no. 1, 195-210 https://doi.org/10.1016/j.jalgebra.2005.04.025
  10. D. E. Dobbs, Coherence, ascent of going-down, and pseudo-valuation domains, Houston J. Math. 4 (1978), no. 4, 551-567
  11. D. E. Dobbs and M. Fontana, Locally pseudovaluation domains, Ann. Mat. Pura Appl. (4) 134 (1983), 147-168 https://doi.org/10.1007/BF01773503
  12. 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
  13. D. E. Dobbs, E. G. 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
  14. M. Fontana, S. Gabelli, and E. G. Houston, UMT-domains and domains with Pr¨ufer integral closure, Comm. Algebra 26 (1998), no. 4, 1017-1039 https://doi.org/10.1080/00927879808826181
  15. M. Fontana and K. A. Loper, Nagata rings, Kronecker function rings, and related semistar operations, Comm. Algebra 31 (2003), no. 10, 4775-4805 https://doi.org/10.1081/AGB-120023132
  16. R. Gilmer, Multiplicative Ideal Theory, Pure and Applied Mathematics, No. 12. Marcel Dekker, Inc., New York, 1972
  17. M. Griffin, Some results on v-multiplication rings, Canad. J. Math. 19 (1967), 710-722 https://doi.org/10.4153/CJM-1967-065-8
  18. J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains, Pacific J. Math. 75 (1978), no. 1, 137-147 https://doi.org/10.2140/pjm.1978.75.137
  19. J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains. II, Houston J. Math. 4 (1978), no. 2, 199-207
  20. E. G. Houston and M. Zafrullah, On t-invertibility. II, Comm. Algebra 17 (1989), no. 8, 1955-1969 https://doi.org/10.1080/00927878908823829
  21. J. A. Huckaba, Commutative Rings with Zero Divisors, Monographs and Textbooks in Pure and Applied Mathematics, 117. Marcel Dekker, Inc., New York, 1988
  22. 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
  23. I. Kaplansky, Commutative Rings, Revised edition. The University of Chicago Press, Chicago, Ill.-London, 1974
  24. H. Kim and Y. S. Park, Some remarks on pseudo-Krull domains, Comm. Algebra 33 (2005), no. 6, 1745-1751 https://doi.org/10.1081/AGB-200063361
  25. A. Okabe and K. Yoshida, Note on strong pseudo-valuation domains, Bull. Fac. Sci. Ibaraki Univ. Ser. A No. 21 (1989), 9-12
  26. J. Querre, Sur une propiete des anneaux de Krull, Bull. Sci. Math. (2) 95 (1971), 341-354
  27. F. Wang, On induced operations and UMT-domains, Sichuan Shifan Daxue Xuebao Ziran Kexue Ban 27 (2004), no. 1, 1-9
  28. M. Zafrullah, Putting t-invertibility to use, Non-Noetherian commutative ring theory, 429-457, Math. Appl., 520, Kluwer Acad. Publ., Dordrecht, 2000

Cited by

  1. ON ALMOST PSEUDO-VALUATION DOMAINS, II vol.19, pp.4, 2011, https://doi.org/10.11568/kjm.2011.19.4.343
  2. PrÜfer-Like Domains and the Nagata Ring of Integral Domains vol.39, pp.11, 2011, https://doi.org/10.1080/00927872.2010.522640
  3. *-NOETHERIAN DOMAINS AND THE RING D[X]N*, II vol.48, pp.1, 2011, https://doi.org/10.4134/JKMS.2011.48.1.049
  4. Graded integral domains which are UMT-domains vol.46, pp.6, 2018, https://doi.org/10.1080/00927872.2017.1399406