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Characterizing rings by a direct decomposition property of their modules

  • Dinh Van Huynh (a1) and S. Tariq Rizvi (a2)
Abstract

A module M is said to satisfy the condition (℘*) if M is a direct sum of a projective module and a quasi-continuous module. In an earlier paper, we described the structure of rings over which every (countably generated) right module satisfies (℘*), and it was shown that such a ring is right artinian. In this note some additional properties of these rings are obtained. Among other results, we show that a ring over which all right modules satisfy (℘*) is also left artinian, but the property (℘*) is not left-right symmetric.

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References
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[1]Anderson, F. W. and Fuller, K. R., Rings and categories of modules, GTM 13, 2nd edition (Springer, New York, 1992).
[2]Dung, N. V., Huynh, D. V., Smith, P. F. and Wisbauer, R., Extending modules (Pitman, London, 1994).
[3]Goodearl, K. R., Singular torsion and the splitting properties, Mem. Amer. Math. Soc. 124 (Amer. Math. Soc., Providence, RI, 1972).
[4]Huyhn, D. V.Structure of some noetherian SI rings’, J. Algebra 254 (2002), 362374.
[5]Huynh, D. V. and Rizvi, S. T., ‘On some classes of artinian rings’, J. Algebra 223 (2000), 133153.
[6]Ivanov, G., ‘Non-local rings whose ideals are quasi-injective’, Bull. Austral. Math. Soc. 6 (1972), 4552.
[7]Lam, T. Y., Lectures on modules and rings, GTM 189 (Springer, New York, 1999).
[8]Wisbauer, R., Foundations of module and ring theory (Gordon and Breach, Reading, 1991).
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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