Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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w-INJECTIVE MODULES AND w-SEMI-HEREDITARY RINGS

  • Wang, Fanggui (College of Mathematics Sichuan Normal University) ;
  • Kim, Hwankoo (School of Computer and Information Engineering Hoseo University)
  • Received : 2013.08.28
  • Published : 2014.05.01
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Abstract

Let R be a commutative ring with identity. An R-module M is said to be w-projective if $Ext\frac{1}{R}$(M,N) is GV-torsion for any torsion-free w-module N. In this paper, we define a ring R to be w-semi-hereditary if every finite type ideal of R is w-projective. To characterize w-semi-hereditary rings, we introduce the concept of w-injective modules and study some basic properties of w-injective modules. Using these concepts, we show that R is w-semi-hereditary if and only if the total quotient ring T(R) of R is a von Neumann regular ring and $R_m$ is a valuation domain for any maximal w-ideal m of R. It is also shown that a connected ring R is w-semi-hereditary if and only if R is a Pr$\ddot{u}$fer v-multiplication domain.

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References

  1. S. El Baghdadi, H. Kim, and F. Wang, Injective-like modules over Krull domains, preprint.
  2. S. Endo, On semi-hereditary rings, J. Math. Soc. Japan 13 (1961), 109-119. https://doi.org/10.2969/jmsj/01320109
  3. R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker Inc., 1972.
  4. S. Glaz, Commutative coherent rings, Lecture Notes in Mathematics, vol. 1371. Springer, Berlin, 1989.
  5. S. Glaz and W. V. Vasconcelos, Flat ideals. II, Manuscripta Math. 22 (1977), no. 4, 325-341. https://doi.org/10.1007/BF01168220
  6. J. S. Golan, Localizations of Noncommutative Rings, Marcel Dekker, New York, 1975.
  7. R. Griffin, Prufer rings with zero divisors, J. Reine Anger. Math. 239 (1969), 55-67.
  8. H. Kim and F. Wang, On LCM-stable modules, J. Algebra Appl. 13 (2014), 1350133, 18 pages.
  9. T. G. Lucas, Strong Prufer rings and the ring of finite fractions, J. Pure Appl. Algebra 84 (1993), no. 1, 59-71. https://doi.org/10.1016/0022-4049(93)90162-M
  10. T. G. Lucas, The Mori property in rings with zero divisors, in Rings, modules, algebras, and abelian groups, 379-400, Lecture Notes in Pure and Appl. Math., 236, Dekker, New York, 2004.
  11. T. G. Lucas, Krull rings, Prufer v-multiplication rings and the ring of finite fractions, Rocky Mountain J. Math. 35 (2005), no. 4, 1251-1326. https://doi.org/10.1216/rmjm/1181069687
  12. J. Marot, Sur le anneaux universellement Japonais, Ph.D. Thesis, Universite de Paris-Sud, 1977.
  13. R. Matsuda, Notes on Prufer v-multiplication rings, Bull. Fac. Sci. Ibaraki Univ. Ser. A 12 (1980), 9-15. https://doi.org/10.5036/bfsiu1968.12.9
  14. J. J. Rotman, An Introduction to Homological Algebra, Academic Press, 1979.
  15. J. R. Silvester, Introduction to Algebraic K-Theory, Chapman and Hall, 1981.
  16. F. Wang, w-projective modules and w-flat modules, Algebra Colloq. 4 (1997), no. 1, 111-120.
  17. F. Wang, w-modules over a PVMD, Proc. ISTAEM, Hong Kong, 117-120, 2001.
  18. F. Wang, Commutative Rings and Star-Operation Theory, (in Chinese), Science Press, Beijing, 2006.
  19. F. Wang, Finitely presented type modules and w-coherent rings, J. Sichuan Normal Univ. 33 (2010), 1-9.
  20. F. Wang and H. Kim, Two generalizations of projective modules and their applications, submitted.
  21. F. Wang and J. Zhang, Injective modules over w-Noetherian rings, Acta Math. Sinica (Chin. Ser.) 53 (2010), no. 6, 1119-1130.
  22. 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
  23. S. Q. Zhao, F. Wang, and H. L. Chen, Flat modules over a commutative ring are w-modules, (in Chinese), J. Sichuan Normal Univ. 35 (2012), 364-366.

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