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Rings characterised by semiprimitive modules

Published online by Cambridge University Press:  17 April 2009

Yasuyuki Hirano ,
Dinh Van Huynh  and
Jae Keol Park
Yasuyuki Hirano
Affiliation:
Department of MathematicsOkayama UniversityOkayama 700Japan
Dinh Van Huynh
Affiliation:
Institute of MathematicsPO Box 631 BohoHanoiVietnam
Jae Keol Park
Affiliation:
Department of MathematicsBusan National UniversityBusan 609–735 South Korea
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Abstract

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A module M is called a CS-module if every submodule of M is essential in a direct summand of M. It is shown that a ring R is semilocal if and only if every semiprimitive right R-module is CS. Furthermore, it is also shown that the following statements are equivalent for a ring R: (i) R is semiprimary and every right (or left) R-module is injective; (ii) every countably generated semiprimitive right R-module is a direct sum of a projective module and an injective module.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 52 , Issue 1 , August 1995 , pp. 107 - 116
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Anderson, F.W. and Fuller, K.R., Rings and Categories of Modules (Springer-Verlag, Berlin, Heidelberg, New York, 1974).CrossRefGoogle Scholar
[2]Faith, C., Algebra II: Ring theory (Springer-Verlag, Berlin, Heidelberg, New York, 1976).CrossRefGoogle Scholar
[3]Goodearl, K.R., Singular torsion and splitting properties, Amer. Math. Soc. Memoirs 124 (American Mathematical Society, 1972).Google Scholar
[4]Huynh, D.V. and Dan, P., ‘A result on Artinian rings’, Math. Japon. 35 (1990), 699–702.Google Scholar
[5]Huynh, D.V. and Wisbauer, R., ‘A structure theorem for SI-modules’, Glasgow Math. J. 34 (1992), 83–89.CrossRefGoogle Scholar
[6]Huynh, D.V. and Wisbauer, R., ‘Self-projective modules with τ-injective factor modules’, J. Algebra 153 (1992), 13–21.CrossRefGoogle Scholar
[7]Huynh, D.V., Dung, N.V. and Wisbauer, R., ‘On modules with finite uniform and Krull dimension’, Arch. Math. 57 (1991), 122–132.CrossRefGoogle Scholar
[8]Michler, G.O. and Villamayor, O.E., ‘On rings whose simple modules are injective’, J. Algebra 25 (1973), 185–201.CrossRefGoogle Scholar
[9]Mohamed, S.H. and Müller, B.J., Continuous and discrete modules, London Math. Soc. Lecture Note Series 147 (London Mathematical Society, 1990).CrossRefGoogle Scholar
[10]Oshiro, K., ‘Lifting modules, extending modules and their applications to QF-rings’, Hokkaido Math. J. 13 (1984), 310–338.Google Scholar
[11]Osofsky, B.L., ‘Rings all of whose finitely generated modules are injective’, Pacific J. Math. 14 (1964), 645–650.CrossRefGoogle Scholar
[12]Osofsky, B.L. and Smith, P.F., ‘Cyclic modules whose quotients have all complement sub-modules direct summands’, J. Algebra 139 (1991), 342–354.CrossRefGoogle Scholar
[13]Wisbauer, R., Foundations of module and ring theory (Gordon and Breach, Reading, 1991).Google Scholar