WEAK GORENSTEIN GLOBAL DIMENSION

Year 2010, Volume: 8 Issue: 8, 140 - 152, 01.12.2010

Driss Bennis

Abstract

In this paper, we investigate the weak Gorenstein global dimension. We are mainly interested in studying the problem when the left and right weak Gorenstein global dimensions coincide. We first show, for GFclosed rings, that the left and right weak Gorenstein global dimensions are equal when they are finite. Then, we prove that the same equality holds for any two-sided coherent ring. We conclude with some examples and a brief discussion of the scope and limits of our results.

Keywords

Gorenstein flat dimension, weak Gorenstein global dimension, weak global dimension, GF-closed rings

References

• D. Bennis, Rings over which the class of Gorenstein flat modules is closed under extensions, Comm. Algebra, 37 (2009), 855–868.
• D. Bennis, A Note on Gorenstein Flat Dimension. Accepted for Publication in Algebra Coll. Available from arXiv:0811.2650v1.
• D. Bennis and N. Mahdou, Global Gorenstein Dimensions, Proc. Amer. Math. Soc., 138 (2010), 461–465.
• L. Bican, R. El Bashir and E. Enochs, All modules have flat covers, Bull. London Math. Soc., 33 (2001), 385–390.
• N. Bourbaki, Alg`ebre Homologique, Chapitre 10, Masson, Paris, 1980.
• T. J. Cheatham and D. R. Stone, Flat and projective character modules, Proc. Amer. Math. Soc., 81 (1981), 175–177.
• J. Chen and N. Ding, Coherent rings with finite self-FP-injective dimension, Comm. Algebra, 24 (1996), 2963–2980.
• L. W. Christensen, Gorenstein Dimensions, Lecture Notes in Math., Springer- Verlag, Berlin, 2000.
• L. W. Christensen, A. Frankild and H. Holm, On Gorenstein projective, injec- tive and flat dimensions - a functorial description with applications, J. Algebra, (2006), 231–279.
• R. R. Colby, On Rings which have flat injective modules, J. Algebra, 35 (1975), –252.
• E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, de Gruyter Expositions in Mathematics, Walter de Gruyter & Co., Berlin, 2000.
• E. E. Enochs and O. M. G. Jenda, Torrecillas, B. Gorenstein flat modules, Nanjing Daxue Xuebao Shuxue Bannian Kan, 10 (1993), 1–9.
• H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189 (2004), 167–193.
• H. Holm, Rings with finite Gorenstein injective dimension, Proc. Amer. Math. Soc., 132 (2004), 1279–1283.
• B. Madox, Absolutely pure modules. Proc. Amer. Math. Soc., 18 (1967), 155–
• J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979.
• B. Stenstr¨om, Coherent rings and FP-injective modules, J. London Math. Soc., (1970), 323–329.
• J. Xu, Flat Covers of Modules, Lecture Notes in Math., Springer-Verlag, Berlin, Driss Bennis Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202,
• University S. M. Ben Abdellah Fez, Morocco, e-mail: driss [e-mail]