Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

TWO GENERALIZATIONS OF LCM-STABLE EXTENSIONS

  • Received : 2012.05.02
  • Published : 2013.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let $R{\subseteq}T$ be an extension of integral domains, X be an indeterminate over T, and R[X] and T[X] be polynomial rings. Then $R{\subseteq}T$ is said to be LCM-stable if $(aR{\cap}bR)T=aT{\cap}bT$ for all $0{\neq}a,b{\in}R$. Let $w_A$ be the so-called $w$-operation on an integral domain A. In this paper, we introduce the notions of $w(e)$- and $w$-LCM-stable extensions: (i) $R{\subseteq}T$ is $w(e)$-LCM-stable if $((aR{\cap}bR)T)_{w_T}=aT{\cap}bT$ for all $0{\neq}a,b{\in}R$ and (ii) $R{\subseteq}T$ is $w$-LCM-stable if $((aR{\cap}bR)T)_{w_R}=(aT{\cap}bT)_{w_R}$ for all $0{\neq}a,b{\in}R$. We prove that LCM-stable extensions are both $w(e)$-LCM-stable and $w$-LCM-stable. We also generalize some results on LCM-stable extensions. Among other things, we show that if R is a Krull domain (resp., $P{\upsilon}MD$), then $R{\subseteq}T$ is $w(e)$-LCM-stable (resp., $w$-LCM-stable) if and only if $R[X]{\subseteq}T[X]$ is $w(e)$-LCM-stable (resp., $w$-LCM-stable).

Keywords

Acknowledgement

Supported by : University of Incheon, NRF

References

  1. T. Akiba, LCM-stableness, Q-stableness and flatness, Kobe J. Math. 2 (1985), no. 1, 67-70.
  2. D. D. Anderson and S. J. Cook, Two star-operations and their induced lattices, Comm. Algebra 28 (2000), no. 5, 2461-2475. https://doi.org/10.1080/00927870008826970
  3. D. D. Anderson, E. G. Houston, and M. Zafrullah, t-linked extensions, the t-class group, and Nagata's theorem, J. Pure Appl. Algebra 86 (1993), no. 2, 109-124. https://doi.org/10.1016/0022-4049(93)90097-D
  4. D. F. Anderson and G. W. Chang, Overrings as intersections of localizations of an integral domain, preprint.
  5. G. W. Chang, *-Noetherian domains and the ring $D[X]N_*$, J. Algebra 297 (2006), no. 1, 216-233. https://doi.org/10.1016/j.jalgebra.2005.08.020
  6. J. T. Condo, LCM-stability of power series extensions characterizes Dedekind domains, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2333-2341.
  7. D. E. Dobbs, On the criteria of D. D. Anderson for invertible and flat ideals, Canad. Math. Bull. 29 (1986), no. 1, 25-32. https://doi.org/10.4153/CMB-1986-004-4
  8. 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
  9. R. Gilmer, An embedding theorem for HCF-rings, Proc. Cambridge Philos. Soc. 68 (1970), 583-587. https://doi.org/10.1017/S0305004100076568
  10. R. Gilmer, Finite element factorization in group rings, Ring theory, 47-61, Lecture Notes in Pure and Appl. Math., Vol. 7, Dekker, New York, 1974.
  11. R. Gilmer, Multiplicative Ideal Theory, Queen's Papers in Pure Appl. Math. 90, Queen's University, Kingston, Ontario, 1992.
  12. J. R. Hedstrom and E. G. Houston, Some remarks on star-operations, J. Pure Appl. Algebra 18 (1980), no. 1, 37-44. https://doi.org/10.1016/0022-4049(80)90114-0
  13. E. G. Houston and M. Zafrullah, On t-invertibility. II, Comm. Algebra 17 (1989), no. 8, 1955-1969. https://doi.org/10.1080/00927878908823829
  14. B. G. Kang, *-operations on integral domains, Ph.D. Dissertation, Univ. Iowa 1987.
  15. 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
  16. D. J. Kwak and Y. S. Park, On t-flat overrings, Chinese J. Math. 23 (1995), no. 1, 17-24.
  17. A. Mimouni, Integral domains in which each ideal is a w-ideal, Comm. Algebra 33 (2005), no. 5, 1345-1355. https://doi.org/10.1081/AGB-200058369
  18. S. Oda and K. Yoshida, Remarks on LCM-stableness and reflexiveness, Math. J. Toyama Univ. 17 (1994), 93-114.
  19. J. Sato and K. Yoshida, The LCM-stability on polynomial extensions, Math. Rep. Toyama Univ. 10 (1987), 75-84.
  20. H. Uda, LCM-stableness in ring extensions, Hiroshima Math. J. 13 (1983), no. 2, 357-377.
  21. H. Uda , $G_2$-stableness and LCM-stableness, HiroshimaMath. J. 18 (1988), no. 1, 47-52.
  22. F. Wang and R. L. McCasland, On w-modules over strong Mori domains, Comm. Algebra 25 (1997), no. 4, 1285-1306. https://doi.org/10.1080/00927879708825920
  23. H. Yin, F. Wang, X. Zhu, and Y. Chen, w-modules over commutative rings, J. Korean Math. Soc. 48 (2011), no. 1, 207-222. https://doi.org/10.4134/JKMS.2011.48.1.207
  24. M. Zafrullah, Putting t-invertibility to use, Non-Noetherian commutative ring theory, 429-457, Math. Appl., 520, Kluwer Acad. Publ., Dordrecht, 2000.

Cited by

  1. ON LCM-STABLE MODULES vol.13, pp.04, 2014, https://doi.org/10.1142/S0219498813501338