The Krull Dimension Theorem Of Module

In this article we use the definition of The Krull dimension of module asIn the second chapter we introduce some definitions and theorem which areneeded in the following chapters.Such as Support,Primary ideal,Ascending chaincondition,Descending chain condition,Local ring,Localization,Norther local ring,Nakayama theorem and Artin-Rees Lemma.In the third chapter we introduce some theories about Northerian local ring,andimprove the dimension theorem of ring. The reference [1] proves the equivalent re-lation of three integers, in this chapter we add another equivalent integer.The most important chapter is the fourth one.First,in the proposition 4.2,weprove three properties of Support,and these properties play an important role in thefollowing proof of theories. Similar to the dimension of ring, the Krull dimensiontheorem of module, the least number of generating element which generate theprimary ideal of a module, the minimum t make M/(x₁,···,x_t) as Artin moduleare equal. The final result Proposition 4.10 is an application of The Krull dimensiontheorem of modules.