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論文名稱(中文) 諾特環上的有限生成模的極限深度的存在性以及它的某些性質
論文名稱(英文) The existence and some properties of the limit depth of a finitely generated module over a Noetherian ring
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 107
學期 2
出版年 108
研究生(中文) 袁國榮
研究生(英文) Kuo-Jung Yuan
電子信箱 99404044@stud.sju.edu.tw
學號 L16051069
學位類別 碩士
語文別 英文
論文頁數 34頁
口試委員 指導教授-蕭仁傑
口試委員-黃一樵
口試委員-劉容真
中文關鍵字 深度  極限深度  深度函數 
英文關鍵字 Depth  Limit depth  Depth function. 
學科別分類
中文摘要 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。
英文摘要 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.
論文目次 Introduction 1
1 Depth 2
1.1 De finition of depth..............................2
1.2 Some properties of depth.........................8
2 Associated Primes 12
2.1 The fi niteness of the associated primes.........12
2.2 The monotonicity of associated primes...........16
2.3 The asymptotic stability of associated primes...17
3 Limit Depth 21
3.1 The existence of the limit depth................21
3.2 Some properties of the limit depth..............27
References 32
A Appendix 34
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