Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Weak Normality and Strong t-closedness of Generalized Power Series Rings

  • Kim, Hwan-Koo (Department of Information Security, College of Engineering, Hoseo University) ;
  • Kwon, Eun-Ok (Department of Applied Mathematics, Changwon National University) ;
  • Kwon, Tae-In (Department of Applied Mathematics, Changwon National University)
  • Received : 2007.10.24
  • Published : 2008.09.30
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Abstract

For an extension $A\;{\subseteq}\;B$ of commutative rings, we present a sufficient conditio for the ring $[[A^{S,\;\leq}]]$ of generalized power series to be weakly normal (resp., stronglyt-closed) in $[[B^{S,\;\leq}]]$, where (S, $\leq$) be a torsion-free cancellative strictly ordered monoid. As a corollary, it can be applied to the ring of power series in infinitely many indeterminates as well as in finite indeterminates.

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References

  1. D. F. Anderson, D. E. Dobbs, and M. Roitman, When is a power series ring n-root closed, J. Pure Appl. Algebra, 114(1997), 111-131. https://doi.org/10.1016/0022-4049(95)00167-0
  2. A. Benhissi, t-closed rings of formal power series, Arch. Math., 73(1999), 109-113. https://doi.org/10.1007/s000130050374
  3. D. E. Dobbs and M. Roitman, Weak Normalization of power series rings, Canad. Math. Bull., 38(1995), 429-433. https://doi.org/10.4153/CMB-1995-062-0
  4. G. A. Elliott and P. Ribenboim, Fields of generalized power series, Arch. Math., 54(1990), 365-371. https://doi.org/10.1007/BF01189583
  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. H. Kim, On t-closedness of generalized power series rings, J. Pure Appl. Algebra, 166(2002), 277-284. https://doi.org/10.1016/S0022-4049(01)00020-2
  7. H. Kim and Y. S. Park, Krull domains of generalized power series, J. Algebra, 237(2001), 292-301. https://doi.org/10.1006/jabr.2000.8581
  8. R. Matsuda, Note on u-closed semigroup rings, Bull. Austral. Math. Soc., 59(1999), 467-471. https://doi.org/10.1017/S0004972700033153
  9. N. Onoda, T. Sugatani, and K. Yoshida, Local quasinomality and closedness type criteria, Houston J. Math., 11(1985), 247-256.
  10. G. Picavet, Seminormal or t-closed schemes and Rees rings, Algebr. Represent. Theory, 1(1998), 255-309. https://doi.org/10.1023/A:1009921701627
  11. G. Picavet and M. Picavet-L'Hermitte, Morphismes t-clos, Comm. in Algebra, 21(1993), 179-219. https://doi.org/10.1080/00927879208824555
  12. G. Picavet and M. Picavet-L'Hermitte, Anneaux t-clos, Comm. in Algebra, 23(1995), 2643-2677. https://doi.org/10.1080/00927879508825364
  13. M. Picavet-L'Hermitte, Weak normality and t-closedness, Comm. in Algebra, 28(2000), 2395-2422. https://doi.org/10.1080/00927870008826967
  14. P. Ribenboim, Rings of generalized power series: Nilpotent elements, Abh. Math. Sem. Univ. Hamburg, 61(1991), 15-33. https://doi.org/10.1007/BF02950748
  15. P. Ribenboim, Special properties of generalized power series, J. Algebra, 173(1995), 566-586. https://doi.org/10.1006/jabr.1995.1103
  16. H. Yanagihara, On an intrinsic definition of weakly normal rings, Kobe J. Math., 2(1985), 89-98.
  17. H. Yanagihara, Some results on weakly normal ring extensions, J. Math. Soc. Japan, 35(1983), 649-661. https://doi.org/10.2969/jmsj/03540649