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Characterizing rings by a direct decomposition property of their modules

Part of: Modules, bimodules and ideals Chain conditions, growth conditions, and other forms of finiteness

Published online by Cambridge University Press:  09 April 2009

Dinh Van Huynh  and
S. Tariq Rizvi
Dinh Van Huynh
Affiliation:
Department of Mathematics, Ohio University, Athens, Ohio 45701, USA, e-mail: huynh@math.ohiou.edu
S. Tariq Rizvi
Affiliation:
Department of Mathematics, The Ohio State University at Lima, Lima, Ohio 45804, USA, e-mail: rizvi.1@osu.edu
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Abstract

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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.

MSC classification

Secondary: 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation 16P20: Artinian rings and modules 16D50: Injective modules, self-injective rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 3 , June 2006 , pp. 359 - 366
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Anderson, F. W. and Fuller, K. R., Rings and categories of modules, GTM 13, 2nd edition (Springer, New York, 1992).CrossRefGoogle Scholar
[2]Dung, N. V., Huynh, D. V., Smith, P. F. and Wisbauer, R., Extending modules (Pitman, London, 1994).Google Scholar
[3]Goodearl, K. R., Singular torsion and the splitting properties, Mem. Amer. Math. Soc. 124 (Amer. Math. Soc., Providence, RI, 1972).CrossRefGoogle Scholar
[4]Huyhn, D. V. ‘Structure of some noetherian SI rings’, J. Algebra 254 (2002), 362–374.Google Scholar
[5]Huynh, D. V. and Rizvi, S. T., ‘On some classes of artinian rings’, J. Algebra 223 (2000), 133–153.CrossRefGoogle Scholar
[6]Ivanov, G., ‘Non-local rings whose ideals are quasi-injective’, Bull. Austral. Math. Soc. 6 (1972), 45–52.CrossRefGoogle Scholar
[7]Lam, T. Y., Lectures on modules and rings, GTM 189 (Springer, New York, 1999).CrossRefGoogle Scholar
[8]Wisbauer, R., Foundations of module and ring theory (Gordon and Breach, Reading, 1991).Google Scholar