Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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CHARACTERIZING S-FLAT MODULES AND S-VON NEUMANN REGULAR RINGS BY UNIFORMITY

  • Zhang, Xiaolei (School of Mathematics and Statistics Shandong University of Technology)
  • Received : 2021.04.09
  • Accepted : 2022.01.24
  • Published : 2022.05.31
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Abstract

Let R be a ring and S a multiplicative subset of R. An R-module T is called u-S-torsion (u-always abbreviates uniformly) provided that sT = 0 for some s ∈ S. The notion of u-S-exact sequences is also introduced from the viewpoint of uniformity. An R-module F is called u-S-flat provided that the induced sequence 0 → A ⊗R F → B ⊗R F → C ⊗R F → 0 is u-S-exact for any u-S-exact sequence 0 → A → B → C → 0. A ring R is called u-S-von Neumann regular provided there exists an element s ∈ S satisfying that for any a ∈ R there exists r ∈ R such that sα = rα2. We obtain that a ring R is a u-S-von Neumann regular ring if and only if any R-module is u-S-flat. Several properties of u-S-flat modules and u-S-von Neumann regular rings are obtained.

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Acknowledgement

The author was supported by the National Natural Science Foundation of China (No. 12061001).

References

  1. D. D. Anderson and T. Dumitrescu, S-Noetherian rings, Comm. Algebra 30 (2002), no. 9, 4407-4416. https://doi.org/10.1081/AGB-120013328
  2. S. Bazzoni and L. Positselski, S-almost perfect commutative rings, J. Algebra 532 (2019), 323-356. https://doi.org/10.1016/j.jalgebra.2019.05.018
  3. D. Bennis and M. El Hajoui, On S-coherence, J. Korean Math. Soc. 55 (2018), no. 6, 1499-1512. https://doi.org/10.4134/JKMS.j170797
  4. L. Fuchs and L. Salce, Modules over non-Noetherian domains, Mathematical Surveys and Monographs, 84, American Mathematical Society, Providence, RI, 2001. https://doi.org/10.1090/surv/084
  5. S. Glaz, Commutative coherent rings, Lecture Notes in Mathematics, 1371, Springer-Verlag, Berlin, 1989. https://doi.org/10.1007/BFb0084570
  6. H. Kim, M. O. Kim, and J. W. Lim, On S-strong Mori domains, J. Algebra 416 (2014), 314-332. https://doi.org/10.1016/j.jalgebra.2014.06.015
  7. J. W. Lim, A note on S-Noetherian domains, Kyungpook Math. J. 55 (2015), no. 3, 507-514. https://doi.org/10.5666/KMJ.2015.55.3.507
  8. J. W. Lim and D. Y. Oh, S-Noetherian properties on amalgamated algebras along an ideal, J. Pure Appl. Algebra 218 (2014), no. 6, 1075-1080. https://doi.org/10.1016/j.jpaa.2013.11.003
  9. C. Nita, Objets noetheriens par rapport a une sous-categorie epaisse d'une categorie abelienne, Rev. Roumaine Math. Pures Appl. 10 (1965), 1459-1467.
  10. B. Stenstrom, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer-Verlag, New York, 1975.
  11. F. Wang and H. Kim, Foundations of commutative rings and their modules, Algebra and Applications, 22, Springer, Singapore, 2016. https://doi.org/10.1007/978-981-10-3337-7