Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-08T03:26:01.192Z Has data issue: false hasContentIssue false
  • English
  • Français

DING-GRADED MODULES AND GORENSTEIN GR-FLAT MODULES

Part of: Rings and algebras with additional structure Modules, bimodules and ideals

Published online by Cambridge University Press:  02 November 2017

LIXIN MAO
LIXIN MAO*
Affiliation:
Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing 211167, China e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let R be a graded ring. We introduce the concepts of Ding gr-injective and Ding gr-projective R-modules, which are the graded analogues of Ding injective and Ding projective modules. Several characterizations and properties of Ding gr-injective and Ding gr-projective modules are obtained. In addition, we investigate the relationships among Gorenstein gr-flat, Ding gr-injective and Ding gr-projective modules.

MSC classification

Primary: 16W50: Graded rings and modules
Secondary: 16D40: Free, projective, and flat modules and ideals 16D50: Injective modules, self-injective rings
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 2 , May 2018 , pp. 339 - 360
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Asensio, M. J., Lopez Ramos, J. A. and Torrecillas, B., Gorenstein gr-injective and gr-projective modules, Comm. Algebra 26 (1998), 225–240.CrossRefGoogle Scholar
2. Asensio, M. J., Lopez Ramos, J. A. and Torrecillas, B., Gorenstein gr-flat modules, Comm. Algebra 26 (1998), 3195–3209.Google Scholar
3. Asensio, M. J., Lopez Ramos, J. A. and Torrecillas, B., FP-gr-injective modules and gr-FC rings, Algebra Number Theory (Marcel Dekker, Inc., New York, 1999), 1–11.Google Scholar
4. Asensio, M. J., Lopez Ramos, J. A. and Torrecillas, B., Covers and envelopes over gr-Gorenstein rings, J. Algebra 215 (1999), 437–459.CrossRefGoogle Scholar
5. Auslander, M. and Bridger, M., Stable module theory, Mem. Amer. Math. Soc., vol. 94 (American Mathematical Society, Providence, 1969).Google Scholar
6. Crivei, S., Prest, M., and Torrecillas, B., Covers in finitely accessible categories, Proc. Amer. Math. Soc. 138 (2010), 1213–1221.Google Scholar
7. Ding, N. Q., Li, Y. L. and Mao, L. X., Strongly Gorenstein flat modules, J. Aust. Math. Soc. 86 (2009), 323–338.CrossRefGoogle Scholar
8. Enochs, E. E., Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), 189–209.Google Scholar
9. Enochs, E. E. and Jenda, O. M. G., Gorenstein injective and Gorenstein projective modules, Math. Z. 220 (1995), 611–633.Google Scholar
10. Enochs, E. E. and Jenda, O. M. G., Relative homological algebra (Walter de Gruyter, Berlin-New York, 2000).Google Scholar
11. Enochs, E. E., Jenda, O. M. G. and Torrecillas, B., Gorenstein flat modules, Nanjing Daxue Xuebao Shuxue Bannian Kan 10 (1993), 1–9.Google Scholar
12. Enochs, E. E. and López-Ramos, J. A., Gorenstein flat modules (Nova Science Publishers, Inc., New York, 2001).Google Scholar
13. Enochs, E. E. and Oyonarte, L., Covers, envelopes and cotorsion theories (Nova Science Publishers, Inc., New York, 2002).Google Scholar
14. García Rozas, J. R., López-Ramos, J. A. and Torrecillas, B., On the existence of flat covers in R-gr, Comm. Algebra 29 (2001), 3341–3349.Google Scholar
15. Garcia Rozas, J. R. and Torrecillas, B., Preserving and reflecting covers by functors: Applications to graded modules, J. Pure Appl. Algebra 112 (1996), 91–107.Google Scholar
16. Gillespie, J., Cotorsion pairs and degreewise homological model structures, Homology, Homotopy Appl. 10 (2008), 283–304.CrossRefGoogle Scholar
17. Gillespie, J., Model structures on modules over Ding-Chen rings, Homology, Homotopy Appl. 12 (2010), 61–73.Google Scholar
18. Göbel, R. and Trlifaj, J., Approximations and endomorphism algebras of modules (Walter de Gruyter, Berlin-New York, 2006).Google Scholar
19. Hermann, M., Ikeda, S. and Orbanz, U., Equimultiplicity and blowing up (Springer-Verlag, New York-Berlin, 1988).Google Scholar
20. Holm, H., Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), 167–193.Google Scholar
21. Malliavin, M. P., Sur les anneaux de groupes FP self-injectifs. C. R. Acad. Sc. Paris 273 (1971), 88–91.Google Scholar
22. Mao, L. X. and Ding, N. Q., Gorenstein FP-injective and Gorenstein flat modules, J. Algebra Appl. 7 (2008), 491–506.Google Scholar
23. Nastasescu, C., Raianu, S. and Van Oystaeyen, F., Modules graded by G-sets, Math. Z. 203 (1990), 605–627.Google Scholar
24. Nastasescu, C., Van Den Bergh, M. and Van Oystaeyen, F., Separable functors applied to graded rings, J. Algebra 123 (1989), 397–413.Google Scholar
25. Nastasescu, C. and Van Oystaeyen, F., Graded ring theory (North-Holland Publishing Company, Amsterdam, New York, Oxford, 1982).Google Scholar
26. Rotman, J. J., An introduction to homological algebra (Academic Press, New York, 1979).Google Scholar
27. Stenström, B., Coherent rings and FP-injective modules, J. London Math. Soc. 2 (1970), 323–329.Google Scholar
28. Stenström, B., Rings of quotients (Springer-Verlag, Berlin, Heidelberg, New York, 1975).Google Scholar
29. Wisbauer, R., Foundations of module and ring theory (Gordon and Breach, Philadelphia, 1991).Google Scholar
30. Yang, X. Y. and Liu, Z. K., FP-gr-injective modules, Math. J. Okayama Univ. 53 (2011), 83–100.Google Scholar
31. Yang, G., Liu, Z. K. and Liang, L., Ding projective and ding injective modules, Algebra Colloq. 20 (2013), 601–612.Google Scholar