Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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ON ALMOST PSEUDO-VALUATION DOMAINS, II

  • Received : 2011.07.21
  • Accepted : 2011.10.20
  • Published : 2011.12.30
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Abstract

Let D be an integral domain, $D^w$ be the $w$-integral closure of D, X be an indeterminate over D, and $N_v=\{f{\in}D[X]{\mid}c(f)_v=D\}$. In this paper, we introduce the concept of $t$-locally APVD. We show that D is a $t$-locally APVD and a UMT-domain if and only if D is a $t$-locally APVD and $D^w$ is a $PvMD$, if and only if D[X] is a $t$-locally APVD, if and only if $D[X]_{N_v}$ is a locally APVD.

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References

  1. A. Badawi, Pseudo-valuation domains; A survey, Proceeding of the Third Palestinian International Conference on Mathematics, 38-59 (2002), World Scientific Publishing Co., New York/London.
  2. A. Badawi, On pseudo-almost valuation domains, Comm. Algebra 35 (2007), 1167-1181. https://doi.org/10.1080/00927870601141951
  3. A. Badawi and E.G. Houston, Powerful ideals, strongly primary ideals, almost pseudo-valuation domains, and conductive domains, Comm. Algebra 30 (2002), 1591-1606. https://doi.org/10.1081/AGB-120013202
  4. 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
  5. G.W. Chang, Locally pseudo-valuation domain of the form $D[X]_{N_{v}}$, J. Korean Math. Soc. 45 (2008), 1405-1416. https://doi.org/10.4134/JKMS.2008.45.5.1405
  6. G.W. Chang, On almost pseudo-valuation domains, Korean J. Math. 18 (2010), 185-193.
  7. G.W. Chang, H. Nam, and J. Park, Strongly primary ideals, in Arithmetical Properties of Commutative Rings and Monoids, Lecture Notes in Pure and Appl. Math., Chapman and Hall, 241 (2005), 378-386.
  8. G.W. Chang and M. Zafrullah, The w-integral closure of integral domains, J. Algebra 295 (2006), 195-210. https://doi.org/10.1016/j.jalgebra.2005.04.025
  9. D.E. Dobbs and M. Fontana, Locally pseudo-valuation domains, Ann. Mat. Pura Appl.(4) 134 (1983), 147-168. https://doi.org/10.1007/BF01773503
  10. D.E. Dobbs, E.G. 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
  11. M. Fontana, S. Gabelli and E.G. Houston, UMT-domains and domains with Prufer integral closure, Comm. Algebra 26 (1998), 1017-1039. https://doi.org/10.1080/00927879808826181
  12. M. Griffin, Some results on v-multiplication rings, Canad. Math. J. 19 (1967), 710-722. https://doi.org/10.4153/CJM-1967-065-8
  13. J.R. Hedstrom and E.G. Housotn, Pseudo-valuation domains, Pacific J. Math. 75 (1978), 137-147. https://doi.org/10.2140/pjm.1978.75.137
  14. E.G. Houston and M. Zafrullah, On t-invertibility, II, Comm. Algebra 17 (1989), 1955-1969. https://doi.org/10.1080/00927878908823829
  15. B.G. Kang, Prufer v-multiplication domains and the ring $D[X]_{N_{v}}$, J. Algebra 123 (1989), 151-170. https://doi.org/10.1016/0021-8693(89)90040-9