Real Valuations And Higher Level Orderings On A Module

In this paper, the notion of real valuations on a module is at first reviewed, andseveral results on real valations are established. Then, the harmony andcompatibility of valuations with higher level orderings on a module are introduced,and the close interplay between valuations and higher level orderings is investigated.The results in this paper show that the harmony and compatibility of valuations withhigher level orderings on a module are not equivalent, which is a difference betweenhigher level orderings and ordinary orderings (i.e. orderings of level 1). Moreover, itis shown that a valuation on a module is real if and only if it is compatiable wih somehigher level ordering on the module.