Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-07-07T20:33:56.886Z Has data issue: false hasContentIssue false
  • English
  • Français

A characterization of noetherian rings by cyclic modules

Published online by Cambridge University Press:  20 January 2009

Dinh Van Huynh
Dinh Van Huynh
Affiliation:
Institute of MathematicsP.O. Box 631 BohoHanoi, Vietnam
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.

It is shown that a ring R is right noetherian if and only if every cyclic right R-module is injective or a direct sum of a projective module and a noetherian module.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 2 , June 1996 , pp. 253 - 262
Copyright
Copyright © Edinburgh Mathematical Society 1996

References

REFERENCES

1. Anderson, F. W. and Fuller, K. R. Rings and Categories of Modules (Springer-Verlag 1974).CrossRefGoogle Scholar
2. Boyle, A. K. and Goodearl, K. R., Rings over which certain modules are injective, Pacific J. Math. 58(1975), 43–53.Google Scholar
3. Chatters, A. W., A characterization of right noetherian rings, Quart. J. Math. Oxford (2) 33 (1982), 65–69.Google Scholar
4. Chatters, A. W. and Hajarnavis, C. R. Rings with Chain Conditions (Pitman, London 1980).Google Scholar
5. Cozzens, J. H., Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75–79.CrossRefGoogle Scholar
6. Damiano, R. F., A right PCI ring is right noetherian, Proc. Amer. Math. Soc. 77 (1979), 11–14.CrossRefGoogle Scholar
7. Van Huynh, Dinh and Dung, Nguyen V., A characterization of artinian rings, Glasgow Math. J. 30 (1988), 67–73.CrossRefGoogle Scholar
8. Van Huynh, Dinh and Smith, P. F., Some rings characterised by their modules, Comm. Algebra (6) 18 (1990), 1971–1988.Google Scholar
9. Van Huynh, Dinh, Dung, Nguyen V. and Smith, P. F., A characterization of rings with Krull dimension, J. Algebra 132 (1990), 104–112.CrossRefGoogle Scholar
10. Van Huynh, Dinh, Dung, Nguyen V. and Wisbauer, R., On modules with finite uniform and Krull dimension, Arch. Math. 57 (1991), 122–132.CrossRefGoogle Scholar
11. Faith, C., When are proper cyclics injective?, Pacific J. Math. 45 (1973), 97–112.CrossRefGoogle Scholar
12. Faith, C., Algebra II: Ring Theory (Springer-Verlag 1976).Google Scholar
13. Goel, V. K., Jain, S. K. and Singh, S., Rings whose cyclic modules are injective or projective, Proc. Amer. Math. Soc. 53 (1975), 16–18.CrossRefGoogle Scholar
14. Gordon, R. and Robson, J. C. Krull dimension (Mem. Amer. Math. Soc. 123, 1973).Google Scholar
15. Kasch, F., Moduln und Ringe (Teubner, Stuttgart 1977).CrossRefGoogle Scholar
16. Kertész, A. (Editor), Associative Rings, Modules and Radicals, in Proc. Colloquium at Keszthely in 1971 (J. Bolyai Math. Soc. North-Holland Publ. Co., Amsterdam-London-Budapest, 1973).Google Scholar
17. Osofsky, B. L. and Smith, P. F., Cyclic modules whose quotients have complements direct summands, J. Algebra 139 (1991), 342–354.Google Scholar
18. Smith, P. F., Rings characterized by their cyclic modules, Canad. J. Math. 24 (1979), 93–111.CrossRefGoogle Scholar
19. Smith, P. F., Van Huynh, Dinh and Dung, Nguyen V., A characterization of noetherian modules, Quart. J. Math. Oxford 41 (1990), 225–235.CrossRefGoogle Scholar
20. Wisbauer, R., Foundations of Module and Ring Theory (Gordon and Breach, Reading 1991).Google Scholar