대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제55권4호
  • /
  • Pages.1007-1012
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  • 2018
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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ON GENERALIZED KRULL POWER SERIES RINGS

  • Le, Thi Ngoc Giau (Faculty of Mathematics and Statistics Ton Duc Thang University) ;
  • Phan, Thanh Toan (Faculty of Mathematics and Statistics Ton Duc Thang University)
  • 투고 : 2017.03.16
  • 심사 : 2017.09.14
  • 발행 : 2018.07.31
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초록

Let R be an integral domain. We prove that the power series ring R[[X]] is a Krull domain if and only if R[[X]] is a generalized Krull domain and t-dim $R{\leq}1$, which improves a well-known result of Paran and Temkin. As a consequence we show that one of the following statements holds: (1) the concepts "Krull domain" and "generalized Krull domain" are the same in power series rings, (2) there exists a non-t-SFT domain R with t-dim R > 1 such that t-dim R[[X]] = 1.

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참고문헌

  1. J. T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc. 177 (1973), 299-304. https://doi.org/10.1090/S0002-9947-1973-0316451-8
  2. G. W. Chang, B. G. Kang, and P. T. Toan, The Krull dimension of power series rings over almost Dedekind domains, J. Algebra 438 (2015), 170-187. https://doi.org/10.1016/j.jalgebra.2015.05.010
  3. G. W. Chang and D. Y. Oh, When D((X)) and D{{X}} are Prufer domains, J. Pure Appl. Algebra 216 (2012), no. 2, 276-279. https://doi.org/10.1016/j.jpaa.2011.06.009
  4. M. D. Fried and M. Jarden, Field Arithmetic, third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 11, Springer-Verlag, Berlin, 2008.
  5. R. Gilmer, Power series rings over a Krull domain, Pacific J. Math. 29 (1969), 543-549. https://doi.org/10.2140/pjm.1969.29.543
  6. R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, Inc., New York, 1972.
  7. M. Griffin, Some results on v-multiplication rings, Canad. J. Math. 19 (1967), 710-722. https://doi.org/10.4153/CJM-1967-065-8
  8. 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
  9. B. G. Kang, K. A. Loper, T. G. Lucas, M. H. Park, and P. T. Toan, The Krull dimension of power series rings over non-SFT rings, J. Pure Appl. Algebra 217 (2013), no. 2, 254- 258. https://doi.org/10.1016/j.jpaa.2012.06.006
  10. B. G. Kang and M. H. Park, A localization of a power series ring over a valuation domain, J. Pure Appl. Algebra 140 (1999), no. 2, 107-124. https://doi.org/10.1016/S0022-4049(97)00211-9
  11. B. G. Kang and M. H. Park, A note on t-SFT-rings, Comm. Algebra 34 (2006), no. 9, 3153-3165. https://doi.org/10.1080/00927870600639476
  12. B. G. Kang and M. H. Park, Krull-dimension of the power series ring over a nondiscrete valuation domain is uncountable, J. Algebra 378 (2013), 12-21. https://doi.org/10.1016/j.jalgebra.2012.05.017
  13. K. A. Loper and T. G. Lucas, Constructing chains of primes in power series rings, J. Algebra 334 (2011), 175-194. https://doi.org/10.1016/j.jalgebra.2011.03.012
  14. K. A. Loper and T. G. Lucas, Constructing chains of primes in power series rings, II, J. Algebra Appl. 12 (2013), no. 1, 1250123, 30 pp.
  15. E. Paran, Split embedding problems over complete domains, Ann. of Math. (2) 170 (2009), no. 2, 899-914. https://doi.org/10.4007/annals.2009.170.899
  16. E. Paran and M. Temkin, Power series over generalized Krull domains, J. Algebra 323 (2010), no. 2, 546-550. https://doi.org/10.1016/j.jalgebra.2009.08.011
  17. F. Pop, Henselian implies large, Ann. of Math. (2) 172 (2010), no. 3, 2183-2195. https://doi.org/10.4007/annals.2010.172.2183
  18. R. Weissauer, Der Hilbertsche Irreduzibilitatssatz, J. Reine Angew. Math. 334 (1982), 203-220.