DOI QR코드

DOI QR Code

KAPLANSKY-TYPE THEOREMS IN GRADED INTEGRAL DOMAINS

  • CHANG, GYU WHAN ;
  • KIM, HWANKOO ;
  • OH, DONG YEOL
  • Received : 2014.08.29
  • Published : 2015.07.31

Abstract

It is well known that an integral domain D is a UFD if and only if every nonzero prime ideal of D contains a nonzero principal prime. This is the so-called Kaplansky's theorem. In this paper, we give this type of characterizations of a graded PvMD (resp., G-GCD domain, GCD domain, $B{\acute{e}}zout$ domain, valuation domain, Krull domain, ${\pi}$-domain).

Keywords

Kaplansky-type theorem;upper to zero;prime (primary) element;graded PvMD;graded GCD domain;graded G-GCD domain;graded $B{\acute{e}}zout$ domain;graded valuation domain;graded Krull domain;graded ${\pi}$-domain

References

  1. D. D. Anderson and D. F. Anderson, Generalized GCD domains, Comment. Math. Univ. St. Paul. 28 (1979), 215-221.
  2. D. D. Anderson and D. F. Anderson, Divisibility properties of graded domains, Canad. J. Math. 34 (1982), no. 1, 196-215. https://doi.org/10.4153/CJM-1982-013-3
  3. D. D. Anderson and D. F. Anderson, Divisorial ideals and invertible ideals in a graded integral domain, J. Algebra 76 (1982), no. 2, 549-569. https://doi.org/10.1016/0021-8693(82)90232-0
  4. D. D. Anderson, T. Dumitrescu, and M. Zafrullah, Almost splitting sets and AGCD domains, Comm. Algebra 32 (2004), no. 1, 147-158. https://doi.org/10.1081/AGB-120027857
  5. D. D. Anderson and M. Zafrullah, On a theorem of Kaplansky, Boll. Un. Mat. Ital. A (7) 8 (1994), no. 3, 397-402.
  6. D. D. Anderson and M. Zafrullah, On t-invertibility, IV, in Factorization in integral domains (Iowa City, IA, 1996), 221-225, Lecture Notes in Pure and Appl. Math., 189, Dekker, New York, 1997.
  7. D. F. Anderson and G. W. Chang, Homogeneous splitting sets of a graded integral domain, J. Algebra 288 (2005), no. 2, 527-554. https://doi.org/10.1016/j.jalgebra.2005.03.007
  8. D. F. Anderson and G. W. Chang, Almost splitting sets in integral domains II, J. Pure Appl. Algebra 208 (2007), no. 1, 351-359. https://doi.org/10.1016/j.jpaa.2006.01.006
  9. D. F. Anderson and G. W. Chang, Graded integral domains and Nagata rings, J. Algebra 387 (2013), 169-184. https://doi.org/10.1016/j.jalgebra.2013.04.021
  10. D. F. Anderson, G. W. Chang, and J. Park, Generalized weakly factorial domains, Houston J. Math. 29 (2003), no. 1, 1-13.
  11. G. W. Chang, Almost splitting sets in integral domains, J. Pure Appl. Algebra 197 (2005), no. 1-3, 279-292. https://doi.org/10.1016/j.jpaa.2004.08.035
  12. G. W. Chang, B. G. Kang, and J. W. Lim, Prufer v-multiplication domains and related domains of the form D + $D-S[{\Gamma}^*]$, J. Algebra 323 (2010), no. 11, 3124-3133. https://doi.org/10.1016/j.jalgebra.2010.03.010
  13. G. W. Chang and H. Kim, Kaplansky-type theorems II, Kyungpook Math. J. 51 (2011), no. 3, 339-344. https://doi.org/10.5666/KMJ.2011.51.3.339
  14. R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.
  15. R. Gilmer, J. Mott, and M. Zafrullah, t-invertibility and comparability, in Commutative ring theory (Fs, 1992), 141-150, Lecture Notes in Pure and Appl. Math., 153, Dekker, New York, 1994.
  16. E. G. Houston, S. B. Malik, and J. L. Mott, Characterizations of *-multiplication domains, Canad. Math. Bull. 27 (1984), no. 1, 48-52. https://doi.org/10.4153/CMB-1984-007-2
  17. J. L. Johnson, The graded ring R[$X_1$, . . . ,$X_n$], Rocky Mountain J. Math. 9 (1979), no. 3, 415-424. https://doi.org/10.1216/RMJ-1979-9-3-415
  18. B. G. Kang, On the converse of a well-known fact about Krull domains, J. Algebra 124 (1989), no. 2, 284-299. https://doi.org/10.1016/0021-8693(89)90131-2
  19. I. Kaplansky, Commutative Rings, Revised Ed., Univ. of Chicago, Chicago, 1974.
  20. H. Kim, Kaplansky-type theorems, Kyungpook Math. J. 40 (2000), no. 1, 9-16.
  21. D. G. Northcott, Lessons on Rings, Modules, and Multiplicities, Cambridge Univ. Press, Cambridge, 1968.
  22. M. Zafrullah, On finite conductor domains, Manuscripta Math. 24 (1978), no. 2, 191-204. https://doi.org/10.1007/BF01310053
  23. M. Zafrullah, A general theory of almost factoriality, Manuscripta Math. 51 (1985), no. 1-3, 29-62. https://doi.org/10.1007/BF01168346