這篇論文主要的目的是要證明理想J(在一個可交換且有單位元的諾特環中)在M/InM 上的極限深度存在，這裡I 是一個R 中的理想而M 則是一個有限生成的R-模。換句話說，我們想要證明下列極限存在，lim depth_R(J,M/I^nM).事實上，這項工作已經由Ｍ.Brodmann[4]所完成。所以我們只是系統性的列出相關的結果和證明。我們首先在第一章中討論深度的概念以及它的某些性質，然後我們在第二章中研究M/InM 的關聯質理想的集合的漸進穩定性。在最後一章中，我們使用前幾章中所得到的結果來證明J 在M/I^nM上的極限深度的存在性並擴展一些深度的性質到極限的情況中。

The main purpose of this paper is to prove that the limit depth of the ideal J (in a commutative Noetherian ring R with identity) on M/I^nM exist, where I is an ideal in R and M is a finitely generated R-module. In other words, we want to prove the following limit exists, lim depth_R(J; M/I^nM). In fact, this work has been completed by M.Brodmann[4]. So we are just list systematically the related results and proofs. We first discuss the concept and some properties of depth in Section 1,and then we study the asymptotic stability of the set of the associated primes of M/I^nM in Section 2. In the fi nal section, we use the results from the previous sections to prove the existence of the limit depth of J on M/I^nM and extend some properties of depth to the limit case.

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