Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T07:36:29.233Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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