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A Condition for Artinian Rings to be Noetherian

Published online by Cambridge University Press:  20 November 2018

Ichiro Murase
Ichiro Murase*
Affiliation:
Japan Women s University, Mejirodai, Bunkyo-ku, Tokyo 112, Japan
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Throughout this paper the word “Artinian (Noetherian) ring” means an associative ring with minimum (maximum) condition on left ideals. According to C. Hopkins, an Artinian ring is Noetherian if it contains a left or right identity [3, p. 728]. However we shall consider Artinian rings without the assumption of existence of such an identity, and the theorem of Hopkins will be reproved.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 4 , 01 August 1978 , pp. 830 - 837
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Artin, E., Rings with minimum condition (University of Michigan Press, Ann Arbor, 1944).Google Scholar
2. Fuchs, L., Abelian groups (Pergamon Press, London, 1960).Google Scholar
3. Hopkins, C., Rings with minimal condition for left ideals, Ann. of Math. Ifi (1939).Google Scholar
4. Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, Massachusetts, 1966).Google Scholar
5. Tominaga, H., On a theorem of Y. -H. Xu (unpublished, University of Okayama, 1976).Google Scholar
6. Y. -H., Xu, A necessary and sufficient condition for the minimal condition for rings to imply the maximal condition, Acta Math. Sinica 18 (1975) (in Chinese).Google Scholar