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.
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.