Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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SOME EXAMPLES OF ALMOST GCD-DOMAINS

  • Received : 2011.07.23
  • Accepted : 2011.08.25
  • Published : 2011.09.30
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Abstract

Let D be an integral domain, X be an indeterminate over D, and D[X] be the polynomial ring over D. We show that D is an almost weakly factorial PvMD if and only if D + XDS[X] is an integrally closed almost GCD-domain for each (saturated) multiplicative subset S of D, if and only if $D+XD_1[X]$ is an integrally closed almost GCD-domain for any t-linked overring $D_1$ of D, if and only if $D_1+XD_2[X]$ is an integrally closed almost GCD-domain for all t-linked overrings $D_1{\subseteq}D_2$ of D.

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References

  1. S. Alaca and K. S. Williams, Introductory Algebraic Number Theory, Cambridge Univ. Press, Cambridge, 2004.
  2. D. D. Anderson, D.F. Anderson, and M. Zafrullah, Rings between D[X] and K[X], Houston J. Math. 17 (1991), 109-129.
  3. D. D. Anderson, D.F. Anderson, and M. Zafrullah, A generalization of unique factorization, Bollettino U.M.I. (7) 9-A (1995), 401-413.
  4. D. D. Anderson, T. Dumitrescu, and M. Zafrullah, Almost splitting sets and AGCD domains, Comm. Algebra 32 (2004), 147-158. https://doi.org/10.1081/AGB-120027857
  5. D. D. Anderson and L.A. Mahaney, On primary factorization, J. Pure Appl. Algebra 54 (1988), 141-154. https://doi.org/10.1016/0022-4049(88)90026-6
  6. D. D. Anderson, J. Mott and M. Zafrullah, Finite character representations for integral domains, Bollettino U.M.I. 6 (1992), 613-630.
  7. D. D. Anderson and M. Zafrullah, Weakly factorial domains and groups of divisibility, Proc. Amer. Math. Soc. 109 (1990), 907-913. https://doi.org/10.1090/S0002-9939-1990-1021893-7
  8. D. F. Anderson, G.W. Chang, and J. Park, Generalized weakly factorial do- mains, Houston J. Math. 29 (2003), 1-13.
  9. D. F. Anderson, G.W. Chang, and J. Park, Weakly Krull and related domains of the form D +M, A +XB[X], and A +$X^{2}$B[X], Rocky Mountain J. Math. 36 (2006), 1-22. https://doi.org/10.1216/rmjm/1181069485
  10. D. F. Anderson and D. El Abidine, The A + XB[X] and A + XB[[X]] con- structions from GCD-domains, J. Pure Appl. Algebra 159 (2001), 15-24. https://doi.org/10.1016/S0022-4049(00)00066-9
  11. G. W. Chang, Almost splitting sets in integral domains, J. Pure Appl. Algebra 197 (2005), 279-292. https://doi.org/10.1016/j.jpaa.2004.08.035
  12. G. W. Chang, Strong Mori domains and the ring D[X]$N_{v}$ , J. Pure Appl. Algebra 197 (2005), 293-304. https://doi.org/10.1016/j.jpaa.2004.08.036
  13. P. M. Cohn, Bezout rings and their subrings, Proc. Cambridge Phil. Soc. 64 (1968), 251-264. https://doi.org/10.1017/S0305004100042791
  14. D. Dobbs, E. Houston, T. Lucas, and M. Zafrullah, t-linked overrings and Prufer v-multiplication domains, Comm. Algebra 17 (1989), 2835-2852. https://doi.org/10.1080/00927878908823879
  15. D. Dobbs, E. Houston, T. Lucas, and M. Zafrullah, t-linked overrings as inter- sections of localizations, Proc. Amer. Math. Soc. 109 (1990), 637-646.
  16. R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972.
  17. I. Kaplansky, Commutative rings, Revised Ed., Univ. of Chicago, Chicago, 1974.
  18. M. Zafrullah, A general theory of almost factoriality, Manuscripta Math. 51 (1985), 29-62. https://doi.org/10.1007/BF01168346
  19. M. Zafrullah, On a property of pre-Schreier domains, Comm. Algebra 15 (1987), 1895-1920. https://doi.org/10.1080/00927878708823512
  20. M. Zafrullah, Various facets of rings between D[X] and K[X], Comm. Algebra 31 (2003), 2497-2540. https://doi.org/10.1081/AGB-120019008