Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-26T06:37:51.220Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Decomposition of Nonsingular CS-Modules

Published online by Cambridge University Press:  20 November 2018

John Clark  and
Nguyen Viet Dung
John Clark
Affiliation:
Department of Mathematics andStatistics, University of Otago, PO Box 56, Dunedin, New Zealand, e-mail:jclark@maths.otago.ac.nz
Nguyen Viet Dung
Affiliation:
Institute of Mathematics, P.O. Box 631 Bo Ho, Hanoi, Vietnam, e-mail:nvdung@thevinh.ac.vn
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 if M is a nonsingular CS-module with an indecomposable decomposition M = ⊕i∊I Mi, then the family {Mi | i € I} is locally semi-T"- nilpotent. This fact is used to prove that any nonsingular self-generator Σ-CS module is a direct sum of uniserial Noetherian quasi-injective submodules. As an application, we provide a new proof of Goodearl's characterization of non-singular rings over which all nonsingular right modules are projective.

Keywords

Primary: 16D70secondary: 16D4016D5016D80nonsingular moduleCS-moduleindecomposable modulequasi-injective moduleuniserial modulehereditary ring
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 3 , 01 September 1996 , pp. 257 - 265
Copyright
Copyright © Canadian Mathematical Society 1996

References

[AF] Anderson, F. W. and Fuller, K. R., Rings and Categories of Modules, Second edition, Graduate Texts in Math. 13, Springer-Verlag, New York, Heidelberg, Berlin, 1992.Google Scholar
[C] Cateforis, V. C., Two-sidedsemisimple maximal quotient rings, Trans. Amer. Math. Soc. 149(1970), 339– 349.Google Scholar
[CH] Chatters, A. W. and Hajarnavis, C. R., Rings with chain conditions, Research Notes in Math. Series 44, Pitman, London, 1980.Google Scholar
[CW] Clark, J. and Wisbauer, R., Σ-extending modules, J. Pure Appl. Algebra 104(1995), 19–32.Google Scholar
[D1] Dung, N. V., On indecomposable decompositions of CS-modules, J. Austral. Math. Soc. Ser. A, to appear.Google Scholar
[D2] Dung, N. V., On indecomposable decompositions of CS-modules II, J. Pure Appl. Algebra, to appear.Google Scholar
[DHSW] Dung, N. V., Huynh, D. V., Smith, P. F. and Wisbauer, R., Extending Modules, Pitman Research Notes in Math. Series 313, Longman, Harlow, 1994.Google Scholar
[DS] Dung, N. V. and Smith, P. F., Σ-CS modules, Comm. Algebra 22(1994), 83–93.Google Scholar
[GD] Garcia, J. L. and Dung, N. V., Some decomposition properties of injective and pure-injective modules, Osaka J. Math. 31(1994), 95–108.Google Scholar
[GJ] Goel, V. K. and Jain, S. K., π-injective modules and rings whose cyclics are n-injective, Comm. Algebra 6(1978), 59–73.Google Scholar
[Gl] Goodearl, K. R., Singular torsion and the splitting properties, Mem. Amer. Math. Soc. 124(1972).Google Scholar
[G2] Goodearl, K. R., Ring Theory: Nonsingular Rings and Modules, Marcel Dekker, New York, Basel, 1976.Google Scholar
[KM] Kamal, M. A. and Muller, B. J., The structure of extending modules over Noetherian rings, Osaka J. Math. 25(1988), 539–551.Google Scholar
[MM] Mohamed, S. H. and B. J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Note Series 147, Cambridge Univ. Press, Cambridge, 1990.Google Scholar
[O] Oshiro, K., Lifting modules, extending modules and their applications to QF-rings, Hokkaido Math. J. 13(1984), 310–338.Google Scholar
[O1] Osofsky, B. L., Rings all of whose finitely generated modules are injective, Pacific J. Math. 14(1964), 645–650.Google Scholar
[O2] Osofsky, B. L., Homological dimension and cardinality, Trans. Amer. Math. Soc. 151(1970), 641–649.Google Scholar
[S] Stenstrôm, B., Rings of Quotients, Springer-Verlag, Berlin, Heidelberg, New York, 1975.Google Scholar
[T] Tachikawa, H., Quasi-Frobenius Rings and Generalizations. QF-3 and QF-1 Rings, Lecture Notes in Math. 351, Springer-Verlag, Berlin, New York, 1973.Google Scholar
[W] Wisbauer, R., Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991.Google Scholar
[Z-H] Zimmermann-Huisgen, B., Endomorphism rings of s elf-generators, Pacific J. Math. 61(1975), 587–602.Google Scholar