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On ℵα-Noetherian Modules

Published online by Cambridge University Press:  20 November 2018

Aron Simis
Aron Simis*
Affiliation:
Queen's University, Kingston, Ontario
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In this note we define two concepts which can be thought of as a generalization of noetherian concepts.

The main result is as follows (Corollary A): If R is a ring whose countably generated (left) ideals are (left) principal, then R is a (left) principal ideal ring.

This result if obtained, more generally, for any (left) R-module and any regular cardinal ℵα (Corollary 1); a cardinal ℵα is regular whenever W(ℵα) = {ordinals γ | card γ < ℵα} has no cofinal subset of cardinality less than ℵα.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 2 , June 1970 , pp. 245 - 247
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Ribenboim, P., Théorie des valuations, Université de Montréal, 1964.Google Scholar
2. Jensen, Chr. U., Homological dimensions of ℵα-coherent rings, Math. Scand. 20 (1967), 55-60.Google Scholar
3. Ribenboim, P., La conjecture d' Artin sur les equations diophantiennes, Queen's University, 1968.Google Scholar