Bennis, Driss; Mahdou, Najib Global Gorenstein dimensions. (English) Zbl 1205.16007 Proc. Am. Math. Soc. 138, No. 2, 461-465 (2010). The authors prove that the global Gorenstein projective dimension of a ring \(R\) is equal to the global Gorenstein injective dimension of \(R\) and that the global Gorenstein flat dimension of \(R\) is smaller than the common value of the terms of this equality. The main result of this paper is an analog of a classical equality that is used to define the global dimension of a ring. Reviewer: Tong Wenting (Nanjing) Cited in 2 ReviewsCited in 85 Documents MSC: 16E10 Homological dimension in associative algebras 16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras 16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) Keywords:global dimension of rings; weak global dimension; Gorenstein homological dimensions; weak Gorenstein global dimension; projective dimension; injective dimension PDFBibTeX XMLCite \textit{D. Bennis} and \textit{N. Mahdou}, Proc. Am. Math. Soc. 138, No. 2, 461--465 (2010; Zbl 1205.16007) Full Text: DOI arXiv References: [1] Driss Bennis and Najib Mahdou, Strongly Gorenstein projective, injective, and flat modules, J. Pure Appl. Algebra 210 (2007), no. 2, 437 – 445. · Zbl 1118.13014 · doi:10.1016/j.jpaa.2006.10.010 [2] D. Bennis, N. Mahdou and K. Ouarghi, Rings over which all modules are strongly Gorenstein projective. Accepted for publication in Rocky Mountain Journal of Mathematics. Available from arXiv:0712.0127v2. · Zbl 1194.13008 [3] Lars Winther Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000. · Zbl 0965.13010 [4] Edgar E. Enochs and Overtoun M. G. Jenda, Relative homological algebra, De Gruyter Expositions in Mathematics, vol. 30, Walter de Gruyter & Co., Berlin, 2000. · Zbl 0952.13001 [5] Robert M. Fossum, Phillip A. Griffith, and Idun Reiten, Trivial extensions of abelian categories, Lecture Notes in Mathematics, Vol. 456, Springer-Verlag, Berlin-New York, 1975. Homological algebra of trivial extensions of abelian categories with applications to ring theory. · Zbl 0303.18006 [6] Henrik Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), no. 1-3, 167 – 193. · Zbl 1050.16003 · doi:10.1016/j.jpaa.2003.11.007 [7] Henrik Holm, Rings with finite Gorenstein injective dimension, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1279 – 1283. · Zbl 1062.16008 [8] H. Holm, Gorenstein Homological Algebra, Ph.D. thesis, University of Copenhagen, Denmark (2004). · Zbl 1050.16003 [9] Arun Vinayak Jategaonkar, A counter-example in ring theory and homological algebra, J. Algebra 12 (1969), 418 – 440. · Zbl 0185.09401 · doi:10.1016/0021-8693(69)90040-4 [10] C. U. Jensen, On the vanishing of \varprojlim\?\(^{1}\)\?, J. Algebra 15 (1970), 151 – 166. · Zbl 0199.36202 · doi:10.1016/0021-8693(70)90071-2 [11] W. K. Nicholson and M. F. Yousif, Quasi-Frobenius rings, Cambridge Tracts in Mathematics, vol. 158, Cambridge University Press, Cambridge, 2003. · Zbl 1042.16009 [12] Joseph J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. · Zbl 0441.18018 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.