Bennis, Driss; Mahdou, Najib Gorenstein global dimensions and cotorsion dimension of rings. (English) Zbl 1172.13008 Commun. Algebra 37, No. 5, 1709-1718 (2009). The aim of the paper is to generalize a result on the classical homological dimension of commutative rings. They prove that if \(R\) is a coherent ring, then \(\text{cot.D}(R)\leq \text{G-gldim}(R)\leq \text{G-wdim}(R)+\text{cot.D}(R)\), where \(\text{cot.D}(R)\) is the global cotorsion dimension of \(R\), \(\text{G-gldim}(R)\) is the Gorenstein global dimension of \(R\), and \(\text{G-wdim}(R)\) is the Gorenstein weak global dimension of \(R\). In particular, if \(\text{cot.D}(R)=0\) (i.e. \(R\) is perfect), then \(\text{G-wdim}(R)=\text{gldim}(R)\), and if \(\text{G-wdim}(R)=0\) (i.e. \(R\) is an IF-ring), then \(\text{cot.D}(R)=\text{G-gldim}(R)\). The result is used to compute the Gorenstein global dimension of some particular cases of trivial extensions of rings and of group rings. Reviewer: Constantin Năstăsescu (Bucureşti) Cited in 1 ReviewCited in 12 Documents MSC: 13D02 Syzygies, resolutions, complexes and commutative rings 13D05 Homological dimension and commutative rings 13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.) Keywords:cotorsion dimension of modules and rings; Gorenstein dimension of modules; Gorenstein global dimension of rings; \(n\)-perfect rings × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Auslander M., Anneaux de Gorenstein et torsion en algèbre commutative (1967) [2] Auslander M., Memoirs of the Amer. Math. Soc. pp 94– (1969) [3] Bass H., Trans. Amer. Math. Soc. 95 pp 466– (1960) · doi:10.1090/S0002-9947-1960-0157984-8 [4] Bennis D., Houston Journal of Mathematics [5] Benson D. J., Pacific J. Math. 196 pp 45– (2000) · Zbl 1073.20500 · doi:10.2140/pjm.2000.196.45 [6] Cartan H., Homological Algebra (1956) [7] Chen J., Comm. Algebra 24 pp 2963– (1996) · Zbl 0855.16001 · doi:10.1080/00927879608825724 [8] Christensen L. W., Gorenstein Dimensions 1747 (2000) · Zbl 0965.13010 · doi:10.1007/BFb0103980 [9] Christensen L. W., J. Algebra 302 pp 231– (2006) · Zbl 1104.13008 · doi:10.1016/j.jalgebra.2005.12.007 [10] Colby R. R., J. Algebra 35 pp 239– (1975) · Zbl 0306.16015 · doi:10.1016/0021-8693(75)90049-6 [11] Ding , N. , Mao , L. ( 2005 ). The cotorsion dimension of modules and rings. Abelian groups, rings, modules, and homological algebra, Lect. Notes Pure Appl. Math. 249:217–233. Available from Ding’s homepage:http://202.119.34.252/Portals/0/jiaoshizhuye/nqding/nqding.htm [12] Enochs E. E., Comm. Algebra 21 pp 3489– (1993) · Zbl 0783.13011 · doi:10.1080/00927879308824744 [13] Enochs E. E., Math. Z. 220 pp 611– (1995) · Zbl 0845.16005 · doi:10.1007/BF02572634 [14] Enochs E. E., J. Algebra 181 pp 288– (1996) · Zbl 0847.16003 · doi:10.1006/jabr.1996.0121 [15] Enochs E. E., Relative Homological Algebra 30 (2000) · Zbl 0952.13001 · doi:10.1515/9783110803662 [16] Enochs E. E., Nanjing Daxue Xuebao Shuxue Bannian Kan 10 pp 1– (1993) [17] Enochs E. E., Proc. Edinb. Math. Soc. 48 pp 75– (2005) · Zbl 1094.16001 · doi:10.1017/S0013091503001056 [18] Fossum R. M., Trivial Extensions of Abelian Categories 456 (1975) · Zbl 0303.18006 [19] Glaz S., Commutative Coherent Rings 1371 (1989) · Zbl 0745.13004 [20] Holm H., J. Pure Appl. Algebra 189 pp 167– (2004) · Zbl 1050.16003 · doi:10.1016/j.jpaa.2003.11.007 [21] Holm H., Gorenstein Homological Algebra (2004) · Zbl 1050.16003 [22] Jain C., Proc. Amer. Math. Soc. 41 pp 437– (1973) · doi:10.1090/S0002-9939-1973-0323828-9 [23] Jensen C. U., J. Algebra 15 pp 151– (1970) · Zbl 0199.36202 · doi:10.1016/0021-8693(70)90071-2 [24] Stenström B., J. London Math. Soc. 2 pp 323– (1970) · Zbl 0194.06602 · doi:10.1112/jlms/s2-2.2.323 [25] Woods S. M., Proc. Amer. Math. Soc. 27 pp 49– (1971) · doi:10.1090/S0002-9939-1971-0271247-4 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.