Real Valuations and Orderings on a Module

Abstract

The purpose of this article is to introduce the notion of real valuations on modules over commutative rings. These real valuations are characterized by their associated valuation triples, and a necessary and sufficient condition for a module to possess a real valuation is established. Moreover, the close interplay between valuations and orderings on a module is investigated by introducing the compatibility of valuations with orderings. Via the compatibility of valuations with orderings, the reality of valuations on a module also is characterized.

Key Words:

• Associated valuation triple
• Compatibility
• Module
• Ordering
• Real valuation
• Valuation

2000 Mathematics Subject Classification:

• 13A18
• 13F30
• 13J30

ACKNOWLEDGMENT

This work is partially supported by a National Key Basic Research Project of China (Grant No. 2004CB318000) and the National Natural Science Foundation of China.

Notes

Communicated by A. Prestel.