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Rings characterized by their right ideals or cyclic modules

Published online by Cambridge University Press:  20 January 2009

Dinh Van Huynh
Affiliation:
Institute of MathematicsP.O. Box 631 Bo HoHanoi, Vietnam
Nguyen V. Dung
Affiliation:
Department of MathematicsUniversity Of GlasgowGlasgow G12 8QW, Scotland, U.K.
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It is well known that a ring R is semiprime Artinian if and only if every right ideal is an injective right R-module. In this paper we shall be concerned with the following general question: given a ring R all of whose right ideals have a certain property, what implications does this have for the ring R itself? In practice, it is not necessary to insist that all right ideals have the property, usually the maximal or essential right ideals will suffice. On the other hand, Osofsky proved that a ring R is semiprime Artinian if and only if every cyclic right R-module is injective. This leads to the second general question: given a ring R all of whose cyclic right R-modules have a certain property, what can one say about R itself?

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1989

References

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