The Compatibility Between M-Valuations And Higher Level Orderings On Commutative Rings

In this paper, the compatibility between M-valuations and higher level orderings is introduced in the category of commutative rings with unit, and several results on the compatibility between M-valuations and higher level orderings are established. Moreover, the compatibility between M-valuation systems and higher level orderings is also investigated. The results in this paper may be considered as some generalizations of the corresponding results on the compatibility between Manis valuations (or M-valuations) and ordinal orderings (i. e. orderings of level 1).This paper is divided into the three sections as follows:Section 1 is the introduction of this paper. In this section, we describe the current status of the research on the compatibility between valuation and orderings.In section 2, we introduce the notion of the compatibility between M-valuations and higher level orderings, and establish several results on the compatibility between M-valuations and higher level orderings.In section 3, the compatibility between M-valuation systems and higher level orderings is investigated.