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Rings characterised by semiprimitive modules

Published online by Cambridge University Press:  17 April 2009

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
Copyright
Copyright © Australian Mathematical Society 1995

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