Research Article
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Year 2018, Volume 23, Issue 23, 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, Volume 23, Issue 23, 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.

Details

Journal Section Articles
Authors

Zhen ZHANG This is me


Jiaqun WEİ This is me

Publication Date January 11, 2018
Published in Issue Year 2018, Volume 23, Issue 23

Cite

Bibtex @research article { ieja373654, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2018}, volume = {23}, number = {23}, pages = {131 - 142}, doi = {10.24330/ieja.373654}, title = {Gorenstein homological dimensions with respect to a semidualizing module}, key = {cite}, author = {Zhang, Zhen and Wei, Jiaqun} }
APA Zhang, Z. & Wei, J. (2018). Gorenstein homological dimensions with respect to a semidualizing module . International Electronic Journal of Algebra , 23 (23) , 131-142 . DOI: 10.24330/ieja.373654
MLA Zhang, Z. , Wei, J. "Gorenstein homological dimensions with respect to a semidualizing module" . International Electronic Journal of Algebra 23 (2018 ): 131-142 <https://dergipark.org.tr/en/pub/ieja/issue/33727/373654>
Chicago Zhang, Z. , Wei, J. "Gorenstein homological dimensions with respect to a semidualizing module". International Electronic Journal of Algebra 23 (2018 ): 131-142
RIS TY - JOUR T1 - Gorenstein homological dimensions with respect to a semidualizing module AU - ZhenZhang, JiaqunWei Y1 - 2018 PY - 2018 N1 - doi: 10.24330/ieja.373654 DO - 10.24330/ieja.373654 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 131 EP - 142 VL - 23 IS - 23 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.373654 UR - https://doi.org/10.24330/ieja.373654 Y2 - 2022 ER -
EndNote %0 International Electronic Journal of Algebra Gorenstein homological dimensions with respect to a semidualizing module %A Zhen Zhang , Jiaqun Wei %T Gorenstein homological dimensions with respect to a semidualizing module %D 2018 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 23 %N 23 %R doi: 10.24330/ieja.373654 %U 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
AMA Zhang Z. , Wei J. Gorenstein homological dimensions with respect to a semidualizing module. IEJA. 2018; 23(23): 131-142.
Vancouver Zhang Z. , Wei J. Gorenstein homological dimensions with respect to a semidualizing module. International Electronic Journal of Algebra. 2018; 23(23): 131-142.
IEEE Z. Zhang and J. Wei , "Gorenstein homological dimensions with respect to a semidualizing module", International Electronic Journal of Algebra, vol. 23, no. 23, pp. 131-142, Jan. 2018, doi:10.24330/ieja.373654