Research Article
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Year 2018, , 131 - 142, 11.01.2018
https://doi.org/10.24330/ieja.373654

Abstract

References

  • D. Bennis and N. Mahdou, Strongly Gorenstein projective, injective, and flat modules, J. Pure Appl. Algebra, 210(2) (2007), 437-445.
  • D. Bennis and N. Mahdou, Global Gorenstein dimensions, Proc. Amer. Math. Soc., 138(2) (2010), 461-465.
  • I. Emmanouil, On the niteness of Gorenstein homological dimensions, J. Al- gebra, 372 (2012), 376-396.
  • E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics, 3, Walter de Gruyter & Co., Berlin, 2000.
  • R. M. Fossum, P. A. Grith and I. Reiten, Trivial Extensions of Abelian Cate- gories, Homological algebra of trivial extensions of abelian categories with ap- plications to ring theory, Lecture Notes in Mathematics, 456, Springer-Verlag, Berlin-New York, 1975.
  • H.-B. Foxby, Gorenstein modules and related modules, Math. Scand., 31 (1972), 267-284.
  • E. S. Golod, G-dimension and generalized perfect ideals, Trudy Mat. Inst. Steklov., 165 (1984), 62-66.
  • H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189(1-3) (2004), 167-193.
  • H. Holm and P. Jrgensen, Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra, 205(2) (2006), 423-445.
  • H. Holm and D. White, Foxby equivalence over associative rings, J. Math. Kyoto Univ., 47(4) (2007), 781-808.
  • S. Sather-Wagstaff, T. Sharif and D. White, Tate cohomology with respect to semidualizing modules, J. Algebra, 324(9) (2010), 2336-2368.
  • W. V. Vasconcelos, Divisor Theory in Module Categories, North-Holland Mathematics Studies, 14, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1974.
  • C. A. Weibel, An Introduction to Homological Algebra, Cambridge Stud. Adv. Math., 38, Cambridge University Press, Cambridge, 1994.
  • D. White, Gorenstein projective dimension with respect to a semidualizing mod- ule, J. Commut. Algebra, 2(1) (2010), 111-137.
  • G. Zhao and J. Sun, Global dimensions of rings with respect to a semidualizing module, avilable from https://arxiv.org/abs/1307.0628.

Gorenstein homological dimensions with respect to a semidualizing module

Year 2018, , 131 - 142, 11.01.2018
https://doi.org/10.24330/ieja.373654

Abstract

In this paper, let R be a commutative ring and C a semidualizing
module. We investigate the (weak) C-Gorenstein global dimension of R
and we get a simple formula to compute the C-Gorenstein global dimension.
Moreover, we compare it with the classical (weak) global dimension of R and
get the relations between them. At last, we compare the weak C-Gorenstein
global dimension with the C-Gorenstein global dimension and we get that they
are equal when R is Noetherian.

References

  • D. Bennis and N. Mahdou, Strongly Gorenstein projective, injective, and flat modules, J. Pure Appl. Algebra, 210(2) (2007), 437-445.
  • D. Bennis and N. Mahdou, Global Gorenstein dimensions, Proc. Amer. Math. Soc., 138(2) (2010), 461-465.
  • I. Emmanouil, On the niteness of Gorenstein homological dimensions, J. Al- gebra, 372 (2012), 376-396.
  • E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics, 3, Walter de Gruyter & Co., Berlin, 2000.
  • R. M. Fossum, P. A. Grith and I. Reiten, Trivial Extensions of Abelian Cate- gories, Homological algebra of trivial extensions of abelian categories with ap- plications to ring theory, Lecture Notes in Mathematics, 456, Springer-Verlag, Berlin-New York, 1975.
  • H.-B. Foxby, Gorenstein modules and related modules, Math. Scand., 31 (1972), 267-284.
  • E. S. Golod, G-dimension and generalized perfect ideals, Trudy Mat. Inst. Steklov., 165 (1984), 62-66.
  • H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189(1-3) (2004), 167-193.
  • H. Holm and P. Jrgensen, Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra, 205(2) (2006), 423-445.
  • H. Holm and D. White, Foxby equivalence over associative rings, J. Math. Kyoto Univ., 47(4) (2007), 781-808.
  • S. Sather-Wagstaff, T. Sharif and D. White, Tate cohomology with respect to semidualizing modules, J. Algebra, 324(9) (2010), 2336-2368.
  • W. V. Vasconcelos, Divisor Theory in Module Categories, North-Holland Mathematics Studies, 14, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1974.
  • C. A. Weibel, An Introduction to Homological Algebra, Cambridge Stud. Adv. Math., 38, Cambridge University Press, Cambridge, 1994.
  • D. White, Gorenstein projective dimension with respect to a semidualizing mod- ule, J. Commut. Algebra, 2(1) (2010), 111-137.
  • G. Zhao and J. Sun, Global dimensions of rings with respect to a semidualizing module, avilable from https://arxiv.org/abs/1307.0628.
There are 15 citations in total.

Details

Journal Section Articles
Authors

Zhen Zhang This is me

Jiaqun Wei This is me

Publication Date January 11, 2018
Published in Issue Year 2018

Cite

APA Zhang, Z., & Wei, J. (2018). Gorenstein homological dimensions with respect to a semidualizing module. International Electronic Journal of Algebra, 23(23), 131-142. https://doi.org/10.24330/ieja.373654
AMA Zhang Z, Wei J. Gorenstein homological dimensions with respect to a semidualizing module. IEJA. January 2018;23(23):131-142. doi:10.24330/ieja.373654
Chicago Zhang, Zhen, and Jiaqun Wei. “Gorenstein Homological Dimensions With Respect to a Semidualizing Module”. International Electronic Journal of Algebra 23, no. 23 (January 2018): 131-42. https://doi.org/10.24330/ieja.373654.
EndNote Zhang Z, Wei J (January 1, 2018) Gorenstein homological dimensions with respect to a semidualizing module. International Electronic Journal of Algebra 23 23 131–142.
IEEE Z. Zhang and J. Wei, “Gorenstein homological dimensions with respect to a semidualizing module”, IEJA, vol. 23, no. 23, pp. 131–142, 2018, doi: 10.24330/ieja.373654.
ISNAD Zhang, Zhen - Wei, Jiaqun. “Gorenstein Homological Dimensions With Respect to a Semidualizing Module”. International Electronic Journal of Algebra 23/23 (January 2018), 131-142. https://doi.org/10.24330/ieja.373654.
JAMA Zhang Z, Wei J. Gorenstein homological dimensions with respect to a semidualizing module. IEJA. 2018;23:131–142.
MLA Zhang, Zhen and Jiaqun Wei. “Gorenstein Homological Dimensions With Respect to a Semidualizing Module”. International Electronic Journal of Algebra, vol. 23, no. 23, 2018, pp. 131-42, doi:10.24330/ieja.373654.
Vancouver Zhang Z, Wei J. Gorenstein homological dimensions with respect to a semidualizing module. IEJA. 2018;23(23):131-42.