SavedAdd your journal

Journals / International Electronic Journal of Algebra / 2024 / Volume: 36 Issue: 36

Strongly J-n-Coherent rings

Zhanmin Zhu(Jiaxing University)
Pages
—
Publication Date
2024-07-12
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Using the concepts of strongly $J$-$n$-injectivity and strongly $J$-$n$-flatness of modules, we also present some characterizations of strongly $J$-$n$-coherent rings and $J$-$n$-semihereditary rings.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Let $R$ be a ring and $n$ a fixed positive integer.
📊
Key Findings & Data:A right $R$-module $M$ is called strongly $J$-$n$-injective if every $R$-homomorphism from an $n$-generated small submodule of a free right $R$-module $F$ to $M$ extends to a homomorphism of $F$ to $M$; a right $R$-module $V$ is said to be strongly $J$-$n$-flat, if for every $n$-generated small submodule $T$ of a free left $R$-module $F$, the canonical map $V\otimes T\rightarrow V\otimes F$ is monic; a ring $R$ is called left strongly $J$-$n$-coherent if every $n$-generated small submodule of a free left $R$-module is finitely presented; a ring $R$ is said to be left $J$-$n$-semihereditary if every $n$-generated small left ideal of $R$ is projective.
💡
Conclusion & Impact:Using the concepts of strongly $J$-$n$-injectivity and strongly $J$-$n$-flatness of modules, we also present some characterizations of strongly $J$-$n$-coherent rings and $J$-$n$-semihereditary rings.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Strongly $J$-$n$-injective module · strongly $J$-$n$-flat module · strongly $J$-$n$-coherent ring

Abstract

Let $R$ be a ring and $n$ a fixed positive integer. A right $R$-module $M$ is called strongly $J$-$n$-injective if every $R$-homomorphism from an $n$-generated small submodule of a free right $R$-module $F$ to $M$ extends to a homomorphism of $F$ to $M$; a right $R$-module $V$ is said to be strongly $J$-$n$-flat, if for every $n$-generated small submodule $T$ of a free left $R$-module $F$, the canonical map $V\otimes T\rightarrow V\otimes F$ is monic; a ring $R$ is called left strongly $J$-$n$-coherent if every $n$-generated small submodule of a free left $R$-module is finitely presented; a ring $R$ is said to be left $J$-$n$-semihereditary if every $n$-generated small left ideal of $R$ is projective. We study strongly $J$-$n$-injective modules, strongly $J$-$n$-flat modules and left strongly $J$-$n$-coherent rings. Using the concepts of strongly $J$-$n$-injectivity and strongly $J$-$n$-flatness of modules, we also present some characterizations of strongly $J$-$n$-coherent rings and $J$-$n$-semihereditary rings.

🔖

Cite This Article

Zhu, Z. (2024). Strongly J-n-Coherent rings. International Electronic Journal of Algebra, Volume: 36 Issue: 36. https://doi.org/10.24330/ieja.1411161

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (22)

  1. H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, NJ, 1956.
  2. S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc., 97 (1960), 457-473.
  3. T. J. Cheatham and D. R. Stone, Flat and projective character modules, Proc. Amer. Math. Soc., 81 (1981), 175-177.
  4. J. L. Chen and N. Q. Ding, A note on existence of envelopes and covers, Bull. Austral. Math. Soc., 54 (1996), 383-390.
  5. J. L. Chen and N. Q. Ding, On n-coherent rings, Comm. Algebra, 24 (1996), 3211-3216.
  6. N. Q. Ding, Y. L. Li and L. X. Mao, J-coherent rings, J. Algebra Appl., 8 (2009), 139-155.
  7. D. D. Dobbs, On n-flat modules over a commutative ring, Bull. Austral. Math. Soc., 43 (1991), 491-498.
  8. E. E. Enochs, A note on absolutely pure modules, Canad. Math. Bull., 19 (1976), 361-362.
  9. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter & Co., Berlin, 2000.
  10. H. Holm and P. Jørgensen, Covers, precovers, and purity, Illinois. J. Math., 52 (2008), 691-703.
  11. W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge University Press, Cambridge, 2003.
  12. J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York-London, 1979.
  13. A. Shamsuddin, n-injective and n-flat modules, Comm. Algebra, 29 (2001), 2039-2050.
  14. B. Stenström, Coherent rings and FP-injective modules, J. London. Math. Soc., 2 (1970), 323-329.
  15. B. Stenstr¨om, Rings of Quotients, Springer-Verlag, New York-Heidelberg, 1975.
  16. R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.
  17. X. X. Zhang, J. L. Chen and J. Zhang, On (m, n)-injective modules and (m, n)- coherent rings, Algebra Colloq., 12 (2005), 149-160.
  18. X. X. Zhang and J. L. Chen, On n-semihereditary and n-coherent rings, Int. Electron. J. Algebra, 1 (2007), 1-10.
  19. Z. M. Zhu and Z. S. Tan, On n-semihereditary rings, Sci. Math. Jpn., 62 (2005), 455-459.
  20. Z. M. Zhu, I-n-coherent rings, I-n-semihereditary rings, and I-regular rings, Ukrainian Math. J., 66 (2014), 857-883.
  21. Z. M. Zhu, I-pure submodules, I-FP-injective modules and I-flat modules, Br. J. Math. Comput. Sci., 8 (2015), 170-188.
  22. Z. M. Zhu, Strongly n-coherent rings, Chinese Ann. Math. Ser. A, 38 (2017), 313-326.