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A characterization of artinian rings

Published online by Cambridge University Press:  18 May 2009

Dinh van Huynh  and
Nguyen V. Dung
Dinh van Huynh
Affiliation:
Institute of Mathematics, P.O. Box 631 Bo Hô, Hanoi-Vietnam
Nguyen V. Dung
Affiliation:
Institute of Mathematics, P.O. Box 631 Bo Hô, Hanoi-Vietnam
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Throughout this paper we consider associative rings with identity and assume that all modules are unitary. As is well known, cyclic modules play an important role in ring theory. Many nice properties of rings can be characterized by their cyclic modules, even by their simple modules. See, for example, [2], [3], [6], [7], [13], [14], [15], [16], [18], [21]. One of the most important results in this direction is the result of Osofsky [14, Theorem] which says: a ring R is semisimple (i.e. right artinian with zero Jacobson radical) if and only if every cyclic right R-module is injective. The other one is due to Vamos [18]: a ring R is right artinian if and only if every cyclic right R-module is finitely embedded.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 30 , Issue 1 , January 1988 , pp. 67 - 73
Copyright
Copyright © Glasgow Mathematical Journal Trust 1988

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