DOI QR코드

DOI QR Code

ON NONNIL-EXACT SEQUENCES AND NONNIL-COMMUTATIVE DIAGRAMS

  • Wei Zhao (School of Mathematics Aba Teachers University) ;
  • Dechuan Zhou (Department of Mathematics Southwest University of Science and Technology)
  • 투고 : 2022.10.10
  • 심사 : 2023.08.01
  • 발행 : 2023.11.01

초록

In this paper, we investigate the nonnil-exact sequences and nonnil-commutative diagrams and show that they behave in a way similar to the classical ones in Abelian categories.

키워드

과제정보

This work was financially supported by the National Natural Science Foundation of China 12061001, 12101515, Sichuan Science and Technology Program 2023NSFSC0074, and Aba Teachers University AS-HBZ2023-34, AS-HBZ2023-35, 20210103002, 20212001128.

참고문헌

  1. D. F. Anderson and A. R. Badawi, On ϕ-Prufer rings and ϕ-Bezout rings, Houston J. Math. 30 (2004), no. 2, 331-343. 
  2. D. F. Anderson and A. R. 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. R. 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. R. Badawi, On Φ-pseudo-valuation rings. II, Houston J. Math. 26 (2000), no. 3, 473-480. 
  6. A. R. Badawi, On ϕ-chained rings and ϕ-pseudo-valuation rings, Houston J. Math. 27 (2001), no. 4, 725-736. 
  7. A. R. Badawi, On divided rings and ϕ-pseudo-valuation rings, in Commutative rings, 5-14, Nova Sci. Publ., Hauppauge, NY, 2002. 
  8. A. R. Badawi, On nonnil-Noetherian rings, Comm. Algebra 31 (2003), no. 4, 1669-1677. https://doi.org/10.1081/AGB-120018502 
  9. A. R. Badawi, On rings with divided nil ideal: a survey, in Commutative algebra and its applications, 21-40, Walter de Gruyter, Berlin, 2009. 
  10. A. R. Badawi and D. E. Dobbs, Strong ring extensions and ϕ-pseudo-valuation rings, Houston J. Math. 32 (2006), no. 2, 379-398. 
  11. A. R. 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. 
  12. A. R. 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 
  13. A. R. Badawi and T. G. Lucas, On Φ-Mori rings, Houston J. Math. 32 (2006), no. 1, 1-32. 
  14. A. Benhissi, Nonnil-Noetherian rings and formal power series, Algebra Colloq. 27 (2020), no. 3, 361-368. https://doi.org/10.1142/S1005386720000292 
  15. 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 
  16. 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 
  17. R. E. 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. E. 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. H. Kim and F. Wang, On ϕ-strong Mori rings, Houston J. Math. 38 (2012), no. 2, 359-371. 
  20. 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 
  21. X. Yang and Z. K. Liu, On nonnil-Noetherian rings, Southeast Asian Bull. Math. 33 (2009), no. 6, 1215-1223. 
  22. X. Zhang, W. Zhao, and F. Wang, On ϕ-flat cotorsion theory J. Guangxi Normal University 39 (2021), no. 2, 119-124. 
  23. W. Zhao, A few kinds of commutative rings with the nilradical as prime ideal, J. Neijiang Normal University 30 (2015), no. 6, 11-14. 
  24. 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 
  25. W. Zhao, On ϕ-exact sequences and ϕ-projective modules, J. Korean Math. Soc. 58 (2021), no. 6, 1513-1528. https://doi.org/10.4134/JKMS.j210180 
  26. W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Comm. Algebra 48 (2022), no. 5, 3079-3090.  https://doi.org/10.1080/00927872.2020.1729362
  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