Characterizations of some rings with 𝒞 -projective, 𝒞 -(FP)-injective and 𝒞 -flat modules

Xiao Guang Yan; Xiao Sheng Zhu

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 3, page 641-652
  • ISSN: 0011-4642

Abstract

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Let R be a commutative ring and 𝒞 a semidualizing R -module. We investigate the relations between 𝒞 -flat modules and 𝒞 -FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings.

How to cite

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Yan, Xiao Guang, and Zhu, Xiao Sheng. "Characterizations of some rings with $\mathcal {C}$-projective, $\mathcal {C}$-(FP)-injective and $\mathcal {C}$-flat modules." Czechoslovak Mathematical Journal 61.3 (2011): 641-652. <http://eudml.org/doc/196347>.

@article{Yan2011,
abstract = {Let $R$ be a commutative ring and $\mathcal \{C\}$ a semidualizing $R$-module. We investigate the relations between $\mathcal \{C\}$-flat modules and $\mathcal \{C\}$-FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings.},
author = {Yan, Xiao Guang, Zhu, Xiao Sheng},
journal = {Czechoslovak Mathematical Journal},
keywords = {semidualizing module; $\mathcal \{C\}$-projective module; $\mathcal \{C\}$-(FP)-injective module; $\mathcal \{C\}$-flat module; noetherian ring; coherent ring; semidualizing module; -projective module; -(FP)-injective module; -flat module; noetherian ring; coherent ring},
language = {eng},
number = {3},
pages = {641-652},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Characterizations of some rings with $\mathcal \{C\}$-projective, $\mathcal \{C\}$-(FP)-injective and $\mathcal \{C\}$-flat modules},
url = {http://eudml.org/doc/196347},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Yan, Xiao Guang
AU - Zhu, Xiao Sheng
TI - Characterizations of some rings with $\mathcal {C}$-projective, $\mathcal {C}$-(FP)-injective and $\mathcal {C}$-flat modules
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 3
SP - 641
EP - 652
AB - Let $R$ be a commutative ring and $\mathcal {C}$ a semidualizing $R$-module. We investigate the relations between $\mathcal {C}$-flat modules and $\mathcal {C}$-FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings.
LA - eng
KW - semidualizing module; $\mathcal {C}$-projective module; $\mathcal {C}$-(FP)-injective module; $\mathcal {C}$-flat module; noetherian ring; coherent ring; semidualizing module; -projective module; -(FP)-injective module; -flat module; noetherian ring; coherent ring
UR - http://eudml.org/doc/196347
ER -

References

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  1. Anderson, F. W., Fuller, K. R., 10.1007/978-1-4684-9913-1_2, Graduate Texts in Mathematics vol. 13 New York-Heidelberg-Berlin: Springer-Verlag (1974). (1974) Zbl0301.16001MR0417223DOI10.1007/978-1-4684-9913-1_2
  2. Avramov, L. L., Foxby, H. B., 10.1112/S0024611597000348, Proc. Lond. Math. Soc., III. Ser. 75 (1997), 241-270. (1997) Zbl0901.13011MR1455856DOI10.1112/S0024611597000348
  3. Chase, S. U., 10.1090/S0002-9947-1960-0120260-3, Trans. Am. Math. Soc. 97 (1961), 457-473. (1961) Zbl0100.26602MR0120260DOI10.1090/S0002-9947-1960-0120260-3
  4. Cheatham, T. J., Stone, D. R., 10.1090/S0002-9939-1981-0593450-2, Proc. Am. Math. Soc. 81 (1981), 175-177. (1981) Zbl0458.16014MR0593450DOI10.1090/S0002-9939-1981-0593450-2
  5. Christensen, L. W., 10.1007/BFb0103984, Lecture Notes in Mathematics, vol. 1747, Springer, Berlin (2000). (2000) Zbl0965.13010MR1799866DOI10.1007/BFb0103984
  6. Christensen, L. W., 10.1090/S0002-9947-01-02627-7, Trans. Am. Math. Soc. 353 (2001), 1839-1883. (2001) Zbl0969.13006MR1813596DOI10.1090/S0002-9947-01-02627-7
  7. Enochs, E. E., Jenda, O. M. G., Xu, J. Z., 10.1090/S0002-9947-96-01624-8, Trans. Am. Math. Soc. 348 (1996), 3223-3234. (1996) Zbl0862.13004MR1355071DOI10.1090/S0002-9947-96-01624-8
  8. Enochs, E. E., Jenda, O. M. G., Relative homological algebra, De Gruyter Expositions in Mathematics vol. 30. Walter de Gruyter, Berlin (2000). (2000) Zbl0952.13001MR1753146
  9. Fieldhouse, D. J., 10.1007/BF02566844, Comment. Math. Helv. 46 (1971), 274-276. (1971) Zbl0219.16017MR0294408DOI10.1007/BF02566844
  10. Foxby, H. B., 10.7146/math.scand.a-11434, Math. Scand. 31 (1972), 267-284. (1972) MR0327752DOI10.7146/math.scand.a-11434
  11. Glaz, S., 10.1007/BFb0084576, (1989). (1989) MR0999133DOI10.1007/BFb0084576
  12. Golod, E. S., G-dimension and generalized perfect ideals, Proc. Steklov Inst. Math. 165 (1985), 67-71. (1985) Zbl0589.13005MR0752933
  13. Holm, H., 10.1016/j.jpaa.2003.11.007, J. Pure Appl. Algebra 189 (2004), 167-193. (2004) Zbl1050.16003MR2038564DOI10.1016/j.jpaa.2003.11.007
  14. Holm, H., Jørgensen, P., 10.1016/j.jpaa.2005.07.010, J. Pure Appl. Algebra. 205 (2006), 423-445. (2006) MR2203625DOI10.1016/j.jpaa.2005.07.010
  15. Holm, H., White, D., 10.1215/kjm/1250692289, J. Math. Kyoto Univ. 47 (2007), 781-808. (2007) Zbl1154.16007MR2413065DOI10.1215/kjm/1250692289
  16. Lam, T. Y., Lectures on Modules and Rings, Graduate Texts in Mathematics 189, Springer-Verlag, New York (1999). (1999) MR1653294
  17. Lambek, J., 10.4153/CMB-1964-021-9, Can. Math. Bull. 7 (1964), 237-243. (1964) Zbl0119.27601MR0163942DOI10.4153/CMB-1964-021-9
  18. Megibben, C., 10.1090/S0002-9939-1970-0294409-8, Proc. Am. Math. Soc. 26 (1970), 561-566. (1970) Zbl0911.16001MR0294409DOI10.1090/S0002-9939-1970-0294409-8
  19. Rotman, J. J., An Introduction to Homological Algebra, Pure and Applied Mathematics vol. 85, Academic Press, New York (1979). (1979) Zbl0441.18018MR0538169
  20. Takahashi, R., White, D., 10.7146/math.scand.a-15121, Math. Scand. 106 (2010), 5-22. (2010) Zbl1193.13012MR2603458DOI10.7146/math.scand.a-15121
  21. Vasconcelos, W. V., Divisor Theory in Module Categories, North-Holland Mathematics Studies, II. Ser. vol. 14, Notes on Mathematica, North-Holland, Amsterdam (1974). (1974) Zbl0296.13005MR0498530
  22. White, D., 10.1216/JCA-2010-2-1-111, J. Commutative Algebra. 2 (2010), 111-137. (2010) MR2607104DOI10.1216/JCA-2010-2-1-111
  23. Sather-Wagstaff, S., Sharif, T., White, D., AB-contexts and stability for Gorenstein flat modules with respect to semidualizing modules, (to appear) in Algebr. Represent. Theor. MR2785915
  24. Zhu, X. S., Characterize rings with character modules, Acta Math. Sinica (Chin. Ser.) 39 (1996), 743-750. (1996) MR1443018
  25. Zhu, X. S., Coherent rings and IF rings, Acta Math. Sin. (Chin. Ser.) 40 (1997), 845-852. (1997) Zbl0899.16004MR1612597

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