Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T08:37:30.977Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Decomposition of Continuous Modules

Published online by Cambridge University Press:  20 November 2018

Bruno J. Müller  and
S. Tariq Rizvi
Bruno J. Müller
Affiliation:
McMaster University, Hamilton, Ontario, Canada
S. Tariq Rizvi
Affiliation:
University of Waterloo, Waterloo, Ontario
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.

We prove two theorems on continuous modules: Decomposition Theorem. A continuous module M has a decomposition, M = M1 ⊕ M2, such that M1 is essential over a direct sum of indecomposable summands Ai of M, and M2 has no uniform submodules; and these data are uniquely determined by M up to isomorphism. Direct Sum Theorem. A finite direct sum of indecomposable modules Ai is continuous if and only if each Ai is continuous and Aj-injective for all j ≠ i.

Keywords

16A52
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 3 , 01 September 1982 , pp. 296 - 301
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Azumaya, G., Mbuntum, F. and Varadarajan, K., On M-projective and M-injective modules, Pacific J. Math. 59 (1975), 9-16.Google Scholar
2. Goel, V. K. and Jain, S. K., π-injective modules and rings whose cyclics are ir-injective, Comm. Algebra 6 (1978), 59-73.Google Scholar
3. Jeremy, L., Modules et anneaux quasi-continus, Canad. Math. Bull. 17 (1974), 217-228.Google Scholar
4. Matlis, E., Injective modules over noetherian rings, Pacifi. J. Math. 8 (1958), 514-528.Google Scholar
5. Mohamed, S. and Bouhy, T., Continuous modules, Arab. J. Sci. Eng. 2 (1977), 107-112.Google Scholar
6. Mohamed, S. and Müller, B. J., Decomposition of dual continuous modules, Lecture Notes in Math. 700, 87-94, Springer 1979.Google Scholar
7. Mohamed, S. and Singh, S., Generalizations for decomposition theorems known over perfect rings, J. Austral. Math. Soc. 24 (1977), 496-510.Google Scholar
8. Papp, Z., On algebraically closed modules, Publ. Math. Decrece. 6 (1959), 311-327.Google Scholar
9. Gordon, R. and Robson, J. C., Krull Dimension, Amer. Math. Soc. Memoirs 133 (1973).Google Scholar
10. Stenström, B., Rings of Quotients, Springer 1975.Google Scholar
11. Utumi, Y., On continuous rings and self injective rings, Trans. Amer. Math. Soc. 118 (1965), 158-173.Google Scholar
12. Warfield, R. B., Decomposition of injective modules, Pacifi. J. Math. 31 (1969), 263-276.Google Scholar
13. Warfield, R. B., Exchange rings and decompositions of modules, Math. Ann. 199 (1972), 31-36.Google Scholar