M-Valuations,Orderings And Preorderings On Commutative Rings

In 2002,Degen Zhang introduced the notion of M-valuations on commutativerings.Both Manis valuations and formally finite V-valuations may be considered asspecial M-valuations.It should be a significant attempt to generalize the relevantresults on Manis valuations and formally finite V-valuations to M-valuations in thecategory of commutative rings.This paper is divided into the five sections as follows:Section 1 is the introduction of this paper.In this section,we describe the currentstatus of the research on various valuations of commutative rings.In Section 2,we give a slightly modified notion of M-valuations which isequivanlent to the notion introduced by Degen Zhang.Moreover,some basic resultsare established.In Section 3,the notion of coveting indeces is introduced.Based on coveringindeces,some important properties of ordered monoid are obtained.In Section 4,the compatibility between M-valuations and orderings is introducedin the category of commutative rings.With the aid of covering indeces,thecompatibility between M-valuations and orderings is characterized.Section 5 deals with the compatibility between M-valuations and preorderingsfor commutative rings.By introducing the so-called T-covering indeces,we obtain anecessary and sufficient condition for a M-valuation to be compatible with apreordering of a commutative ring.