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Certain Artinian Rings are Noetherian

Published online by Cambridge University Press:  20 November 2018

Robert C. Shock*
Affiliation:
Southern Illinois University, Carbondale, Illinois
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Throughout this paper the word “ring” will mean an associative ring which need not have an identity element. There are Artinian rings which are not Noetherian, for example C(p) with zero multiplication. These are the only such rings in that an Artinian ring R is Noetherian if and only if R contains no subgroups of type C(p) [1, p. 285]. However, a certain class of Artinian rings is Noetherian. A famous theorem of C. Hopkins states that an Artinian ring with an identity element is Noetherian [3, p. 69]. The proofs of these theorems involve the method of “factoring through the nilpotent Jacobson radical of the ring”. In this paper we state necessary and sufficient conditions for an Artinian ring (and an Artinian module) to be Noetherian. Our proof avoids the concept of the Jacobson radical and depends primarily upon the concept of the length of a composition series. As a corollary we obtain the result of Hopkins.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Fuchs, L., Abelian groups (Pergamon Press, London, 1960).Google Scholar
2. Kaplansky, I., Infinite abelian groups (University of Michigan Press, Ann Arbor, 1954).Google Scholar
3. Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, Massachusetts, 1966).Google Scholar
4. Shock, R., Dual generalizations of the Artinian and Noetherian conditions (to appear).Google Scholar