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I-n-Coherent Rings, I-n-Semihereditary Rings, and I-Regular Rings

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Ukrainian Mathematical Journal Aims and scope

Let R be a ring, let I be an ideal of R, and let n be a fixed positive integer. We define and study I-n-injective modules and I-n-flat modules. Moreover, we define and study left I-n-coherent rings, left I-n-semihereditary rings, and I-regular rings. By using the concepts of I-n-injectivity and I-n-flatness of modules, we also present some characterizations of the left I-n-coherent rings, left I-n-semihereditary rings, and I-regular rings.

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References

  1. S. Ahmad, “n-Injective and n-flat modules,” Comm. Algebra, 29, 2039–2050 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  2. I. Amin, M. Yousif, and N. Zeyada, “Soc-injective rings and modules,” Comm. Algebra, 33, 4229–4250 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  3. T. J. Cheatham and D. R. Stone, “Flat and projective character modules,” Proc. Amer. Math. Soc., 81, 175–177 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  4. J. L. Chen and N. Q. Ding, “A note on existence of envelopes and covers,” Bull. Austral. Math. Soc., 54, 383–390 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  5. J. L. Chen and N. Q. Ding, “On n-coherent rings,” Comm. Algebra, 24, 3211–3216 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  6. J. L. Chen, N. Q. Ding, Y. L. Li, and Y. Q. Zhou, “On (m, n)-injectivity of modules,” Comm. Algebra, 29, 5589–5603 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  7. F. Couchot, “Flat modules over valuation rings,” J. Pure Appl. Algebra, 211, 235–247 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  8. N. Q. Ding, Y. L. Li, and L. X. Mao, “J-coherent rings,” J. Algebra Appl., 8, 139–155 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  9. D. D. Dobbs, “On n-flat modules over a commutative ring,” Bull. Austral. Math. Soc., 43, 491–498 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  10. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, de Gruyter, Berlin, New York (2000).

    Book  MATH  Google Scholar 

  11. R. N. Gupta, “On f-injective modules and semihereditary rings,” Proc. Nat. Inst. Sci. India A, 35, 323–328 (1969).

    MATH  Google Scholar 

  12. H. Holm and P. Jørgensen, “Covers, precovers, and purity,” Illinois J. Math., 52, 691–703 (2008).

    MathSciNet  Google Scholar 

  13. S. Jøndrup, “P.p. rings and finitely generated flat ideals,” Proc. Amer. Math. Soc., 28, 431–435 (1971).

    MathSciNet  Google Scholar 

  14. N. Mahdou, “On Costa’s conjecture,” Comm. Algebra, 29, 2775–2785 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  15. L. X. Mao, “On P-coherent endomorphism rings,” Proc. Indian Acad. Sci. Math. Sci., 118, 557–567 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  16. W. K. Nicholson and M. F. Yousif, “Principally injective rings,” J. Algebra, 174, 77–93 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  17. W. K. Nicholson and M. F. Yousif, “Mininjective rings,” J. Algebra, 187, 548–578 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  18. W. K. Nicholson and M. F. Yousif, “Weakly continuous and C2-rings,” Comm. Algebra, 29, 2429–2446 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  19. W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Univ. Press, Cambridge (2003).

    Book  MATH  Google Scholar 

  20. B. Stenström, Rings of Quotients, Springer, Berlin, etc. (1975).

    Book  MATH  Google Scholar 

  21. R. Wisbauer, Foundations of Module and Ring Theory, Gordon & Breach, Reading (1991).

    MATH  Google Scholar 

  22. M. F. Yousif and Y. Q. Zhou, “Rings for which certain elements have the principal extension property,” Algebra Colloq., 10, 501–512 (2003).

    MATH  MathSciNet  Google Scholar 

  23. X. X. Zhang and J. L. Chen, “On (m, n)-injective modules and (m, n)-coherent rings,” Algebra Colloq., 12, 149–160 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  24. X. X. Zhang and J. L. Chen, “On n-semihereditary and n-coherent rings,” Int. Electron. J. Algebra, 1, 1–10 (2007).

    MathSciNet  Google Scholar 

  25. Z. M. Zhu and Z. S. Tan, “On n-semihereditary rings,” Sci. Math. Jap., 62, 455–459 (2005).

    MATH  MathSciNet  Google Scholar 

  26. Z. M. Zhu, “Some results on MP-injectivity and MGP-injectivity of rings and modules,” Ukr. Math. Zh., 63, No. 10, 1426–1433 (2011).

    Google Scholar 

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 6, pp. 767–786, June, 2014.

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Zhanmin, Z. I-n-Coherent Rings, I-n-Semihereditary Rings, and I-Regular Rings. Ukr Math J 66, 857–883 (2014). https://doi.org/10.1007/s11253-014-0978-8

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  • DOI: https://doi.org/10.1007/s11253-014-0978-8

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