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DP-PROJECTIVE MODULES AND DIMENSIONS

Year 2014, , 101 - 116, 01.06.2014
https://doi.org/10.24330/ieja.266241

Abstract

In this paper, we introduce the notion of DP-projective modules.
It is shown that a left R-module M over a ring R is DP-projective if and only if
it is a cokernel of a Ding projective preenvelope f : A → B with B projective.
It is also shown that a ring R is semisimple if and only if every module is
DP-projective. Moreover, we investigate (global) DP-projective dimensions of
modules and rings. It is shown that l.DP-dim(R) = l.DP-ID(R). In addition,
other applications of those dimensions defined in this way are presented.

References

  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd ed. Graduate Texts in Mathematics, 13, New York: Springer-Verlag, 1992.
  • H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc., 95 (1960), 466–488.
  • S. Breaz, Modules M such that ExtR(M, −) commuts with direct limits, Algebra Represent. Theory, (to appear), doi: 10.1007/s10468-012-9382-y.
  • N. Q. Ding and J. L. Chen, The flat dimensions of injective modules, Manuscripta Math., 78 (1993), 165–177.
  • N. Q. Ding, Y. L. Li and L. X. Mao, Strongly Gorenstein flat modules, J. Aust. Math. Soc., 86 (2009), 323–338.
  • E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin, 2000.
  • X. H. Fu, H. Y. Zhu and N. Q. Ding, On copure projective modules and copure projective dimensions, Comm. Algebra, 40 (2012), 343–359.
  • J. Gillespie, Model structures on modules over Ding-Chen rings, Homology, Homotopy Appl., 12(1) (2010), 61–73.
  • T. Ishikawa, On injective modules and flat modules, J. Math. Soc. Japan, 17(3) (1965), 291–296.
  • C. U. Jensen, On the vanishing of lim(i), J. Algebra, 15(2) (1970), 151–166. ←−
  • L. X. Mao and N. Q. Ding, Gorenstein F P -injective and Gorenstein flat mod- ules, J. Algebra Appl., 7(4) (2008), 491–506.
  • N. Mahdou and M. Tamekkante, Strongly Gorenstein flat modules and dimen- sions, Chin. Ann. Math., 32B(4) (2011), 533–548.
  • J. J. Rotman, An Introduction to Homological Algebra, 2nd ed. Springer, 2009.
  • C. H. Yang, Strongly Gorenstein flat and Gorenstein FP-injective modules, Turk. J. Math., 37 (2013), 218–230.
  • G. Yang, Homological properties of modules over Ding-Chen rings, J. Korean Math. Soc., 49 (1) (2012), 31–47.
  • G. Yang, Z. K. Liu and L. Liang, Ding projective and Ding injective modules, Algebra Colloq., 20(4) (2013), 601-612.
  • C. X. Zhang and L. M. Wang, Strongly Gorenstein flat dimensions, J. Math. Research & Exposition, 31(6) (2011), 977–988. Tiwei Zhao
  • School of Mathematics and Computer Science Hubei University Wuhan, Hubei, People’s Republic of China e-mail: tiweizhao@gmail.com
Year 2014, , 101 - 116, 01.06.2014
https://doi.org/10.24330/ieja.266241

Abstract

References

  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd ed. Graduate Texts in Mathematics, 13, New York: Springer-Verlag, 1992.
  • H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc., 95 (1960), 466–488.
  • S. Breaz, Modules M such that ExtR(M, −) commuts with direct limits, Algebra Represent. Theory, (to appear), doi: 10.1007/s10468-012-9382-y.
  • N. Q. Ding and J. L. Chen, The flat dimensions of injective modules, Manuscripta Math., 78 (1993), 165–177.
  • N. Q. Ding, Y. L. Li and L. X. Mao, Strongly Gorenstein flat modules, J. Aust. Math. Soc., 86 (2009), 323–338.
  • E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin, 2000.
  • X. H. Fu, H. Y. Zhu and N. Q. Ding, On copure projective modules and copure projective dimensions, Comm. Algebra, 40 (2012), 343–359.
  • J. Gillespie, Model structures on modules over Ding-Chen rings, Homology, Homotopy Appl., 12(1) (2010), 61–73.
  • T. Ishikawa, On injective modules and flat modules, J. Math. Soc. Japan, 17(3) (1965), 291–296.
  • C. U. Jensen, On the vanishing of lim(i), J. Algebra, 15(2) (1970), 151–166. ←−
  • L. X. Mao and N. Q. Ding, Gorenstein F P -injective and Gorenstein flat mod- ules, J. Algebra Appl., 7(4) (2008), 491–506.
  • N. Mahdou and M. Tamekkante, Strongly Gorenstein flat modules and dimen- sions, Chin. Ann. Math., 32B(4) (2011), 533–548.
  • J. J. Rotman, An Introduction to Homological Algebra, 2nd ed. Springer, 2009.
  • C. H. Yang, Strongly Gorenstein flat and Gorenstein FP-injective modules, Turk. J. Math., 37 (2013), 218–230.
  • G. Yang, Homological properties of modules over Ding-Chen rings, J. Korean Math. Soc., 49 (1) (2012), 31–47.
  • G. Yang, Z. K. Liu and L. Liang, Ding projective and Ding injective modules, Algebra Colloq., 20(4) (2013), 601-612.
  • C. X. Zhang and L. M. Wang, Strongly Gorenstein flat dimensions, J. Math. Research & Exposition, 31(6) (2011), 977–988. Tiwei Zhao
  • School of Mathematics and Computer Science Hubei University Wuhan, Hubei, People’s Republic of China e-mail: tiweizhao@gmail.com
There are 18 citations in total.

Details

Other ID JA49GY48ZU
Journal Section Articles
Authors

Tiwei Zhao This is me

Publication Date June 1, 2014
Published in Issue Year 2014

Cite

APA Zhao, T. (2014). DP-PROJECTIVE MODULES AND DIMENSIONS. International Electronic Journal of Algebra, 15(15), 101-116. https://doi.org/10.24330/ieja.266241
AMA Zhao T. DP-PROJECTIVE MODULES AND DIMENSIONS. IEJA. June 2014;15(15):101-116. doi:10.24330/ieja.266241
Chicago Zhao, Tiwei. “DP-PROJECTIVE MODULES AND DIMENSIONS”. International Electronic Journal of Algebra 15, no. 15 (June 2014): 101-16. https://doi.org/10.24330/ieja.266241.
EndNote Zhao T (June 1, 2014) DP-PROJECTIVE MODULES AND DIMENSIONS. International Electronic Journal of Algebra 15 15 101–116.
IEEE T. Zhao, “DP-PROJECTIVE MODULES AND DIMENSIONS”, IEJA, vol. 15, no. 15, pp. 101–116, 2014, doi: 10.24330/ieja.266241.
ISNAD Zhao, Tiwei. “DP-PROJECTIVE MODULES AND DIMENSIONS”. International Electronic Journal of Algebra 15/15 (June 2014), 101-116. https://doi.org/10.24330/ieja.266241.
JAMA Zhao T. DP-PROJECTIVE MODULES AND DIMENSIONS. IEJA. 2014;15:101–116.
MLA Zhao, Tiwei. “DP-PROJECTIVE MODULES AND DIMENSIONS”. International Electronic Journal of Algebra, vol. 15, no. 15, 2014, pp. 101-16, doi:10.24330/ieja.266241.
Vancouver Zhao T. DP-PROJECTIVE MODULES AND DIMENSIONS. IEJA. 2014;15(15):101-16.