Bulletin of the Korean Mathematical Society (대한수학회보)

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NOTES ON FINITELY GENERATED FLAT MODULES

  • Tarizadeh, Abolfazl (Department of Mathematics Faculty of Basic Sciences University of Maragheh)
  • Received : 2019.03.17
  • Accepted : 2019.09.19
  • Published : 2020.03.31
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Abstract

In this paper, the projectivity of finitely generated flat modules of a commutative ring are studied from a topological point of view. Then various interesting results are obtained. For instance, it is shown that if a ring has either finitely many minimal primes or finitely many maximal ideals then every finitely generated flat module over it is projective. It is also shown that if a particular subset of the prime spectrum of a ring satisfies some certain ascending or descending chain conditions, then every finitely generated flat module over this ring is projective. These results generalize some major results in the literature on the projectivity of finitely generated flat modules.

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Acknowledgement

The author would like to give heartfelt thanks to the referee for very careful reading of the paper and for his/her very valuable comments and suggestions which improved the paper.

References

  1. S. H. Cox, Jr. and R. L. Pendleton, Rings for which certain flat modules are projective, Trans. Amer. Math. Soc. 150 (1970), 139-156. https://doi.org/10.2307/1995487
  2. S. Endo, On flat modules over commutative rings, J. Math. Soc. Japan 14 (1962), 284-291. https://doi.org/10.2969/jmsj/01430284
  3. M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43-60. https://doi.org/10.2307/1995344
  4. S. Jondrup, On finitely generated flat modules, Math. Scand. 26 (1970), 233-240. https://doi.org/10.7146/math.scand.a-10979
  5. A. J. de Jong et al., The stacks project, see http://stacks.math.columbia.edu.
  6. I. Kaplansky, Projective modules, Ann. of Math (2) 68 (1958), 372-377. https://doi.org/10.2307/1970252
  7. T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, 189, Springer-Verlag, New York, 1999. https://doi.org/10.1007/978-1-4612-0525-8
  8. D. Lazard, Disconnexites des spectres d'anneaux et des preschemas, Bull. Soc. Math. France 95 (1967), 95-108. https://doi.org/10.24033/bsmf.1649
  9. H. Matsumura, Commutative Ring Theory, translated from the Japanese by M. Reid, second edition, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1989.
  10. G. Puninski and P. Rothmaler, When every finitely generated flat module is projective, J. Algebra 277 (2004), no. 2, 542-558. https://doi.org/10.1016/j.jalgebra.2003.10.027
  11. J. J. Rotman, An Introduction to Homological Algebra, second edition, Universitext, Springer, New York, 2009. https://doi.org/10.1007/b98977
  12. A. Tarizadeh, On the projectivity of finitely generated flat modules, accepted, appearing in Extracta Mathematicae, arXiv:1701.07735v5 [math.AC]
  13. A. Tarizadeh, Flat topology and its dual aspects, Comm. Algebra 47 (2019), no. 1, 195-205. https://doi.org/10.1080/00927872.2018.1469637
  14. A. Tarizadeh, The upper topology and its relation with the projective modules, submitted, arXiv:1612.05745v4 [math.AC]
  15. W. V. Vasconcelos, On finitely generated flat modules, Trans. Amer. Math. Soc. 138 (1969), 505-512. https://doi.org/10.2307/1994928
  16. W. V. Vasconcelos, On projective modules of finite rank, Proc. Amer. Math. Soc. 22 (1969), 430-433. https://doi.org/10.2307/2037071
  17. R. Wiegand, Globalization theorems for locally finitely generated modules, Pacific J. Math. 39 (1971), 269-274. http://projecteuclid.org/euclid.pjm/1102969790 https://doi.org/10.2140/pjm.1971.39.269