(n, m)-STRONGLY GORENSTEIN PROJECTIVE MODULES

Abstract

This paper is a continuation of the papers J. Pure Appl. Algebra, 210 (2007), 437–445 and J. Algebra Appl., 8 (2009), 219–227. Namely, we introduce and study a doubly filtered set of classes of modules of finite Gorenstein projective dimension, which are called (n, m)-strongly Gorenstein projective ((n, m)-SG-projective for short)(for integers n ≥ 1 and m ≥ 0). We are mainly interested in studying syzygies of these modules. As consequences, we show that a module M has Gorenstein projective dimension at most m if and only if M ⊕ G is (1, m)-SG-projective for some Gorenstein projective module G. And, over rings of finite left finitistic flat dimension, that a module of finite Gorenstein projective dimension has finite projective dimension if and only if it has finite flat dimension.

Keywords

• Gorenstein projective modules, Gorenstein projective dimension, (n-)strongly Gorenstein projective modules, (n, m)-SG-projective modules

References

1. M. Auslander, Anneaux de Gorenstein et torsion en alg`ebre commutative, Secr´etariat math´ematique, Paris, 1967, S´eminaire d’alg`ebre commutative dirig´e par Pierre Samuel, 1966/67. Texte r´edig´e, d’apr`es des expos´es de Mau- rice Auslander, par Marquerite Mangeney, Christian Peskine et Lucien Szpiro,
2. Ecole Normale Superieure de Jeunes Filles. M. Auslander and M. Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I. 1969.
3. D. Bennis and N. Mahdou, Strongly Gorenstein projective, injective, and flat modules, J. Pure Appl. Algebra, 210 (2007), 437–445.
4. D. Bennis and N. Mahdou, A generalization of strongly Gorenstein projective modules, J. Algebra Appl., 8 (2009), 219–227.
5. D. Bennis and N. Mahdou, Global Gorenstein Dimensions. Accepted for pub- lication in Proc. Amer. Math. Soc., Available from arXiv:0611358v4.
6. D. Bennis and N. Mahdou, Global Gorenstein dimensions of polynomial rings and of direct products of rings, Accepted for publication in Houston J. Math. Available from arXiv:0712.0126v2.
7. L. W. Christensen, Gorenstein dimensions, Lecture Notes in Math., Springer- Verlag, Berlin, 2000.
8. E. E. Enochs and O. M. G. Jenda, Relative homological algebra, Walter de Gruyter, Berlin-New York, 2000.

9. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z., 220 (1995), 611–633.
10. E. E. Enochs and O. M. G. Jenda, On Gorenstein injective modules, Comm. Algebra, 21 (1993), 3489–3501.
11. E. Enochs, O. M. G. Jenda and J. Xu, Foxby duality and Gorenstein injective and projective modules, Trans. Amer. Math. Soc., 348 (1996), 3223–3234.
12. H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189 (2004), 167–193.
13. G. Zhao and Z. Huang, n-Strongly Gorenstein Projective, Injective and Flat Modules, Available from arXiv:0904.3045v1.
14. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979. Driss Bennis
15. Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202,
16. University S. M. Ben Abdellah, Fez, Morocco e-mail: driss [e-mail]