M-Valuation Systems Of Commutative Rings

In 2002, Digen Zhang[13]introduced the notion of M-valuations on commutative rings, and obtained many relevant results. In this paper, we introduce the notion of M-valuation systems for a commutative ring, and investigate the M-valuations from the inside of a commutative ring. Some results on M-valuations and M-valuation systems are established.This paper is divided into four sections as follows:Section 1 is an introduction of this paper. In this section, we describe the current status of the research on various valuations of commutative rings.In section 2, we give the notion of M-valuation systems for a commutative ring, and establish some basic results on M-valuation systems.In section 3, based on the results in section 2, the interplay between M-valuations and M-valuation systems is discussed, and the close interrelation between them is established.Section 4 introduces the notion of standardizations of M-valuation systems. The equivalence of a M-valuation system and its standardization is investigated.