Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON 𝜙-EXACT SEQUENCES AND 𝜙-PROJECTIVE MODULES

  • Zhao, Wei (Department of Mathematics Nanjing University and School of Mathematics ABa Teachers University)
  • Received : 2021.03.14
  • Accepted : 2021.07.06
  • Published : 2021.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let R be a commutative ring with prime nilradical Nil(R) and M an R-module. Define the map 𝜙 : R → RNil(R) by ${\phi}(r)=\frac{r}{1}$ for r ∈ R and 𝜓 : M → MNil(R) by ${\psi}(x)=\frac{x}{1}$ for x ∈ M. Then 𝜓(M) is a 𝜙(R)-module. An R-module P is said to be 𝜙-projective if 𝜓(P) is projective as a 𝜙(R)-module. In this paper, 𝜙-exact sequences and 𝜙-projective R-modules are introduced and the rings over which all R-modules are 𝜙-projective are investigated.

Keywords

Acknowledgement

This work was financially supported by the National Natural Science Foundation of China 12061001, 11861001, the China Postdoctoral Science Foundation 2021M691526, the Science and Technology Plan Project of Aba Prefecture 20RKX0001, and Aba Teachers University ASB20-02, ASA20-02, ASC20-02, 201901011, 201907019, 201910107, 201910108.

References

  1. D. F. Anderson and A. Badawi, On ϕ-Prufer rings and ϕ-Bezout rings, Houston J. Math. 30 (2004), no. 2, 331-343.
  2. D. F. Anderson and A. Badawi, On ϕ-Dedekind rings and ϕ-Krull rings, Houston J. Math. 31 (2005), no. 4, 1007-1022.
  3. K. Bacem and B. Ali, Nonnil-coherent rings, Beitr. Algebra Geom. 57 (2016), no. 2, 297-305. https://doi.org/10.1007/s13366-015-0260-8
  4. A. Badawi, On ϕ-pseudo-valuation rings, in Advances in commutative ring theory (Fez, 1997), 101-110, Lecture Notes in Pure and Appl. Math., 205, Dekker, New York, 1999.
  5. A. Badawi, On Φ-pseudo-valuation rings. II, Houston J. Math. 26 (2000), no. 3, 473-480.
  6. A. Badawi, On ϕ-chained rings and ϕ-pseudo-valuation rings, Houston J. Math. 27 (2001), no. 4, 725-736.
  7. A. Badawi, On divided rings and ϕ-pseudo-valuation rings, Commutative rings, 5-14, Nova Sci. Publ., Hauppauge, NY, 2002.
  8. A. Badawi, On nonnil-Noetherian rings, Comm. Algebra 31 (2003), no. 4, 1669-1677. https://doi.org/10.1081/AGB-120018502
  9. A. Badawi, Factoring nonnil ideals into prime and invertible ideals, Bull. London Math. Soc. 37 (2005), no. 5, 665-672. https://doi.org/10.1112/S0024609305004509
  10. A. Badawi, On rings with divided nil ideal: a survey, in Commutative algebra and its applications, 21-40, Walter de Gruyter, Berlin, 2009.
  11. A. Badawi and D. E. Dobbs, Strong ring extensions and ϕ-pseudo-valuation rings, Houston J. Math. 32 (2006), no. 2, 379-398.
  12. A. Badawi and A. Jaballah, Some finiteness conditions on the set of overrings of a ϕ-ring, Houston J. Math. 34 (2008), no. 2, 397-408.
  13. A. Badawi and T. G. Lucas, Rings with prime nilradical, in Arithmetical properties of commutative rings and monoids, 198-212, Lect. Notes Pure Appl. Math., 241, Chapman & Hall/CRC, Boca Raton, FL, 2005. https://doi.org/10.1201/9781420028249.ch11
  14. A. Badawi and T. G. Lucas, On Φ-Mori rings, Houston J. Math. 32 (2006), no. 1, 1-32.
  15. A. Benhissi, Nonnil-Noetherian rings and formal power series, Algebra Colloq. 27 (2020), no. 3, 361-368. https://doi.org/10.1142/S1005386720000292
  16. A. Y. Darani and M. Rahmatinia, On ϕ-Schreier rings, J. Korean Math. Soc. 53 (2016), no. 5, 1057-1075. https://doi.org/10.4134/JKMS.j150380
  17. R. El Khalfaoui and N. Mahdou, The ϕ-Krull dimension of some commutative extensions, Comm. Algebra 48 (2020), no. 9, 3800-3810. https://doi.org/10.1080/00927872.2020.1747479
  18. A. El Khalfi, H. Kim, and N. Mahdou, On ϕ-piecewise Noetherian rings, Comm. Algebra 49 (2021), no. 3, 1324-1337. https://doi.org/10.1080/00927872.2020.1834571
  19. S. Hizem and A. Benhissi, Nonnil-Noetherian rings and the SFT property, Rocky Mountain J. Math. 41 (2011), no. 5, 1483-1500. https://doi.org/10.1216/RMJ-2011-41-5-1483
  20. H. Kim and F. Wang, On ϕ-strong Mori rings, Houston J. Math. 38 (2012), no. 2, 359-371.
  21. C. Lomp and A. Sant'Ana, Comparability, distributivity and non-commutative ϕ-rings, in Groups, rings and group rings, 205-217, Contemp. Math., 499, Amer. Math. Soc., Providence, RI, 2009. https://doi.org/10.1090/conm/499/09804
  22. I. Palmer and J.-E. Roos, Explicit formulae for the global homological dimensions of trivial extensions of rings, J. Algebra 27 (1973), 380-413. https://doi.org/10.1016/0021-8693(73)90113-0
  23. 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
  24. X. Y. Yang and Z. K. Liu, On nonnil-Noetherian rings, Southeast Asian Bull. Math. 33 (2009), no. 6, 1215-1223.
  25. W. Zhao, A few kinds of commutative rings with the nilradical as prime ideal, J. Neijiang Normal University 30 (2015), no. 6, 11-14.
  26. W. Zhao, On Φ-flat modules and Φ-Prufer rings, J. Korean Math. Soc. 55 (2018), no. 5, 1221-1233. https://doi.org/10.4134/JKMS.j170667
  27. W. Zhao, F. Wang, and G. Tang, On ϕ-von Neumann regular rings, J. Korean Math. Soc. 50 (2013), no. 1, 219-229. https://doi.org/10.4134/JKMS.2013.50.1.219
  28. W. Zhao, F. Wang, and X. Zhang, On Φ-projective modules and Φ-Prufer rings, Comm. Algebra 48 (2020), no. 7, 3079-3090. https://doi.org/10.1080/00927872.2020.1729362