Browse > Article
http://dx.doi.org/10.4134/BKMS.b190294

NOTES ON FINITELY GENERATED FLAT MODULES  

Tarizadeh, Abolfazl (Department of Mathematics Faculty of Basic Sciences University of Maragheh)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.2, 2020 , pp. 419-427 More about this Journal
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.
Keywords
Flat module; flat topology; patch topology; projectivity; S-ring;
Citations & Related Records
연도 인용수 순위
  • Reference
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   DOI
2 S. Endo, On flat modules over commutative rings, J. Math. Soc. Japan 14 (1962), 284-291. https://doi.org/10.2969/jmsj/01430284   DOI
3 M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43-60. https://doi.org/10.2307/1995344   DOI
4 S. Jondrup, On finitely generated flat modules, Math. Scand. 26 (1970), 233-240. https://doi.org/10.7146/math.scand.a-10979   DOI
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   DOI
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.   DOI
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   DOI
11 A. Tarizadeh, The upper topology and its relation with the projective modules, submitted, arXiv:1612.05745v4 [math.AC]
12 J. J. Rotman, An Introduction to Homological Algebra, second edition, Universitext, Springer, New York, 2009. https://doi.org/10.1007/b98977
13 A. Tarizadeh, On the projectivity of finitely generated flat modules, accepted, appearing in Extracta Mathematicae, arXiv:1701.07735v5 [math.AC]
14 A. Tarizadeh, Flat topology and its dual aspects, Comm. Algebra 47 (2019), no. 1, 195-205. https://doi.org/10.1080/00927872.2018.1469637   DOI
15 W. V. Vasconcelos, On finitely generated flat modules, Trans. Amer. Math. Soc. 138 (1969), 505-512. https://doi.org/10.2307/1994928   DOI
16 W. V. Vasconcelos, On projective modules of finite rank, Proc. Amer. Math. Soc. 22 (1969), 430-433. https://doi.org/10.2307/2037071   DOI
17 R. Wiegand, Globalization theorems for locally finitely generated modules, Pacific J. Math. 39 (1971), 269-274. http://projecteuclid.org/euclid.pjm/1102969790   DOI