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ON PIECEWISE NOETHERIAN DOMAINS

  • Chang, Gyu Whan (Department of Mathematics Education Incheon National University) ;
  • Kim, Hwankoo (School of Computer and Information Engineering Hoseo University) ;
  • Wang, Fanggui (College of Mathematics and Software Science Sichuan Normal University)
  • Received : 2015.04.01
  • Published : 2016.05.01

Abstract

In this paper, we study piecewise Noetherian (resp., piecewise w-Noetherian) properties in several settings including flat (resp., t-flat) overrings, Nagata rings, integral domains of finite character (resp., w-finite character), pullbacks of a certain type, polynomial rings, and D + XK[X] constructions.

Keywords

References

  1. D. D. Anderson, G. W. Chang, and M. Zafrullah, Integral domains of finite t-character, J. Algebra 396 (2013), 169-183. https://doi.org/10.1016/j.jalgebra.2013.08.014
  2. D. D. Anderson, J. Matijevic, and W. Nichols, The Krull intersection Theorem. II, Pacific J. Math. 66 (1976), no. 1, 15-22. https://doi.org/10.2140/pjm.1976.66.15
  3. J. T. Arnold and J. Brewer, On flat overrings, ideal transforms and generalized transforms of a commutative ring, J. Algebra 18 (1971), 254-264. https://doi.org/10.1016/0021-8693(71)90058-5
  4. J. A. Beachy andW. D.Weakley, Piecewise Noetherian rings, Comm. Algebra 12 (1984), no. 21-22, 2679-2706. https://doi.org/10.1080/00927878408823127
  5. J. A. Beachy andW. D.Weakley, A note on prime ideals which test injectivity, Comm. Algebra 15 (1987), no. 3, 471-478. https://doi.org/10.1080/00927878708823428
  6. A. Benhissi, CCA pour les ideaux radicaux et divisoriels, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 44(92) (2001), no. 2, 119-135.
  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. 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
  9. M. D'Anna, C. A. Finocchiaro, and M. Fontana, Amalgamated algebras along an ideal, in: Commutative algebra and its applications, 155-172, Walter de Gruyter, Berlin, 2009.
  10. 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
  11. S. El Baghdadi and S. Gabelli, Ring-theoretic properties of PvMDs, Comm. Algebra 35 (2007), no. 5, 1607-1625. https://doi.org/10.1080/00927870601169283
  12. S. El Baghdadi, H. Kim, and F. Wang, A note on generalized Krull domains, J. Algebra Appl. 13 (2014), no. 7, 1450029, 18 pp.
  13. A. Facchini, Generalized Dedekind domains and their injective modules, J. Pure Appl. Algebra 94 (1994), no. 2, 159-173. https://doi.org/10.1016/0022-4049(94)90030-2
  14. M. Fontana and S. Gabelli, On the class group and the local class group of a pullback, J. Algebra 181 (1996), no. 3, 803-835. https://doi.org/10.1006/jabr.1996.0147
  15. R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.
  16. E. Houston and M. Zafrullah, On t-invertibility II, Comm. Algebra 17 (1989), no. 8, 1955-1969. https://doi.org/10.1080/00927878908823829
  17. 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
  18. H. Kim and T. I. Kwon, Integral domains which are t-locally Noetherian, J. Chungcheong Math. Soc. 24 (2011), 843-848.
  19. H. Kim, T. I. Kwon, and M. S. Rhee, A note on zero divisors in w-Noetherian-like rings, Bull. Korean Math. Soc. 51 (2014), no. 6, 1851-1861. https://doi.org/10.4134/BKMS.2014.51.6.1851
  20. D. J. Kwak and Y. S. Park, On t-flat overrings, Chinese J. Math. 23 (1995), no. 1, 17-24.
  21. C. P. Lu, Modules satisfying ACC on a certain type of colons, Pacific J. Math. 131 (1988), no. 2, 303-318. https://doi.org/10.2140/pjm.1988.131.303
  22. C. P. Lu, Modules and rings satisfying (accr), Proc. Amer. Math. Soc. 117 (1993), no. 1, 5-10. https://doi.org/10.1090/S0002-9939-1993-1104398-7
  23. J. Ohm and R. Pendleton, Rings with Noetherian spectrum, Duke Math. J. 35 (1968), 631-639. https://doi.org/10.1215/S0012-7094-68-03565-5
  24. M. H. Park, Power series rings over strong Mori domains, J. Algebra 270 (2003), no. 1, 361-368. https://doi.org/10.1016/j.jalgebra.2003.07.009
  25. A. C. Pearson, Noncommutative Piecewise Noetherian Rings, Ph. D. thesis, Northern Illinois University, 2011.
  26. M. Tamekkante, K. Louartiti, and M. Chhiti, Chain conditions in amalgamated algebras along an ideal, Arab. J. Math. (Springer) 2 (2013), no. 4, 403-408. https://doi.org/10.1007/s40065-013-0075-0
  27. F. Wang and R. L. McCasland, On w-modules over strong Mori domains, Comm. Al-gebra 25 (1997), no. 4, 1285-1306. https://doi.org/10.1080/00927879708825920