Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T16:52:11.312Z Has data issue: false hasContentIssue false
  • English
  • Français

Modules whose closed submodules are finitely generated

Published online by Cambridge University Press:  20 January 2009

Nguyen V. Dung
Nguyen V. Dung
Affiliation:
Institute of MathematicsP.O. Box 631 Bo HoHanoi, 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.

A module M is called a CC-module if every closed submodule of M is cyclic. It is shown that a cyclic module M is a direct sum of indecomposable submodules if all quotients of cyclic submodules of M are CC-modules. This theorem generalizes a recent result of B. L. Osofsky and P. F. Smith on cyclic completely CS-modules. Some further applications are given for cyclic modules which are decomposed into projectives and injectives.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 1 , February 1991 , pp. 161 - 166
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

REFERENCES

1.Chatters, A. W. and Hajarnavis, C. R., Rings in which every complement right ideal is a direct summand, Quart. J. Math. Oxford Ser. (2), 28 (1977), 61–80.CrossRefGoogle Scholar
2.Cozzens, J. H. and Faith, C. C., Simple Noetherian Rings (Cambridge University Press, Cambridge, London, New York, 1975).CrossRefGoogle Scholar
3.Damiano, R. F., A right PCI ring is right Noetherian, Proc. Amer. Math. Soc. 77 (1979), 11–14.CrossRefGoogle Scholar
4.Van Huynh, Dinh and Dung, Nguyen V., Rings with restricted injective condition, Arch. Math. (Basel), to appear.Google Scholar
5.Van Huynh, Dinh, Dung, Nguyen V. and Wisbauer, R., Quasi-injective modules with ACC or DCC on essential submodules, Arch. Math. (Basel) 53 (1989), 252–255.Google Scholar
6.Osofsky, B. L., Rings all of whose finitely generated modules are injective, Pacific J. Math. 14 (1964), 645–650.CrossRefGoogle Scholar
7.Osofsky, B. L., Non-injective cyclic modules, Proc. Amer. Math. Soc. 19 (1968), 1383–1384.CrossRefGoogle Scholar
8.Osofsky, B. L. and Smith, P. F., Cyclic modules whose quotients have complements direct summands, preprint.Google Scholar
9.Sharpe, D. W. and Vamos, P., Injective modules (Cambridge University Press, Cambridge, London, New York, 1972).Google Scholar
10.Smith, P. F., Decomposing modules into projectives and injectives, Pacific J. Math. 74 (1978), 247–266.CrossRefGoogle Scholar
11.Smith, P. F., Rings characterized by their cyclic modules, Canad. J. Math. 31 (1979), 93–111.CrossRefGoogle Scholar
12.Smith, P. F., Van Huynh, Dinh and Dung, Nguyen V., A characterization of Noetherian modules, Quart. J. Math. Oxford Ser. (2), to appear.Google Scholar