Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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POWER SERIES RINGS OVER PRÜFER v-MULTIPLICATION DOMAINS

  • Chang, Gyu Whan (Department of Mathematics Education Incheon National University)
  • Received : 2015.03.05
  • Published : 2016.03.01
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Abstract

Let D be an integral domain, {$X_{\alpha}$} be a nonempty set of indeterminates over D, and $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}_1}$ be the first type power series ring over D. We show that if D is a t-SFT $Pr{\ddot{u}}fer$ v-multiplication domain, then $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}}_{1_{D-\{0\}}}$ is a Krull domain, and $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}}_1$ is a $Pr{\ddot{u}}fer$ v-multiplication domain if and only if D is a Krull domain.

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Acknowledgement

Supported by : Incheon National University

References

  1. D. D. Anderson, B. G. Kang, and M. H. Park, Anti-Archimedean rings and power series rings, Comm. Algebra 26 (1998), 3223-3238. https://doi.org/10.1080/00927879808826338
  2. D. D. Anderson and M. Zafrullah, Almost Bezout domains, J. Algebra 142 (1991), 285-309. https://doi.org/10.1016/0021-8693(91)90309-V
  3. J. Arnold, Power series rings over Prufer domains, Pacific J. Math. 44 (1973), 1-11. https://doi.org/10.2140/pjm.1973.44.1
  4. J. Arnold, Power series rings with finite Krull dimension, Indiana Univ. Math. J. 31 (1982), 897-911. https://doi.org/10.1512/iumj.1982.31.31061
  5. G. W. Chang, A pinched-Krull domain at a prime ideal, Comm. Algebra 30 (2002), 3669-3686. https://doi.org/10.1081/AGB-120005812
  6. G. W. Chang, Spectral localizing systems that are t-splitting multiplicative sets of ideals, J. Korean Math. Soc. 44 (2007), 863-872. https://doi.org/10.4134/JKMS.2007.44.4.863
  7. G. W. Chang and M. Fontana, Upper to zero in polynomial rings and Prufer-like domains, Comm. Algebra 37 (2009), 164-192. https://doi.org/10.1080/00927870802243564
  8. G. W. Chang and D. Y. Oh, The rings $D((X))_i$ and D{{X}}$_i$, J. Algebra Appl. 12 (2013), 1250147 (11 pages).
  9. D. Dobbs, E. Houston, T. Lucas, and M. Zafrullah, t-linked overrings and Prufer vmultiplication domains, Comm. Algebra 17 (1989), 2835-2852. https://doi.org/10.1080/00927878908823879
  10. M. Fontana and S. Gabelli, On the class group and the local class group of a pullback, J. Algebra 181 (1996), 803-835. https://doi.org/10.1006/jabr.1996.0147
  11. M. Fontana, S. Gabelli, and E. Houston, UMT-domains and domains with Prufer integral closure, Comm. Algebra 26 (1998), 1017-1039. https://doi.org/10.1080/00927879808826181
  12. 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
  13. R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.
  14. R. Gilmer and W. Heinzer, Primary ideals with finitely generated radical in a commutative ring, Manuscripta Math. 78 (1993), 201-221. https://doi.org/10.1007/BF02599309
  15. E. Houston and M. Zafrullah, On t-invertibility II, Comm. Algebra 17 (1989), 1955-1969. https://doi.org/10.1080/00927878908823829
  16. B. G. Kang, Prufer v-multiplication domains and the ring $R[X]_{N_v}$, J. Algebra 123 (1989), 151-170. https://doi.org/10.1016/0021-8693(89)90040-9
  17. B. G. Kang and M. H. Park, A note on t-SFT-rings, Comm. Algebra 34 (2006), 3153-3165. https://doi.org/10.1080/00927870600639476
  18. D. J. Kwak and Y. S. Park, On t-flat overrings, Chinese J. Math. 23 (1995), 17-24.
  19. J. Mott and M. Zafrullah, On Krull domains, Arch. Math. 56 (1991), 559-568. https://doi.org/10.1007/BF01246772

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