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On the extension of orders in ordered modules

Published online by Cambridge University Press:  17 April 2009

P. Ribenboim
P. Ribenboim
Affiliation:
Queen's University, Kingston, Ontario.
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Abstract

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We introduce the notion of a positively independent set of elements in an ordered module. With this concept we determine a necessary and sufficient condition which insures that on a strictly ordered module over a strictly ordered ring there exists a strict total order refining the given order. This generalizes a previous result of Fuchs, concerning the case of ordered abelian groups.

As an application, let R be a strictly ordered totally ordered ring and let M be the R-module of all mappings from a set I into R, with pointwise order; then this order on M may be refined to a strict total order.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 2 , Issue 1 , February 1970 , pp. 81 - 88
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Fuchs, L., Partially ordered algebraic systems, (Pergamon Press, Oxford, London, New York, Paris, 1963).Google Scholar
[2]Ribenboim, P., “On ordered modules”, J. Reine Angew. Math. 225 (1967), 120–146.Google Scholar