Gorenstein homological dimensions with respect to a semidualizing module

Yıl 2018, , 131 - 142, 11.01.2018

Zhen Zhang Jiaqun Wei

https://doi.org/10.24330/ieja.373654

Cited By: 3

Öz

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.

Anahtar Kelimeler

Semidualizing module, trivial extension ring, C-Gorenstein projective dimension

Kaynakça

• 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.