On Generalized Euclidean Rings

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湯何曼(Abdul Rahman Tambunan) 查詢紙本館藏

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數學系

論文名稱

(On Generalized Euclidean Rings)

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摘要(中)

這篇論文研究由P. M. Cohn 所提出之 generalized Euclidean ring（簡稱 GE-ring）的概念。當中也介紹了GE-ring 與 GE_n-ring（非GE-ring）的例子與其性質。從 Bass 的一項結果：「一個環 R 的 stable rank（表示成 sr(R)）與 GLn(R) 有關」可以得知，所有 stable rank 為 1 的環會是一個 GE-ring。另外有一事實則是一個主理想整環（principal ideal domain）的 stable rank 則會小於等於 2。若一主理想整環的 stable rank 為 1，則其必為一歐幾里得環。文中也給出一些 stable rank 大於 1 之 GE-ring 的例子。對於在二次體（quadratic field） K=(Q√d)（其中d為一無平方數因數 square free）中的所有整數環（ring of integers, O_K），可以得到 sr(O_K)=2。

摘要(英)

In this thesis, a generalized Euclidean ring, or GE-ring for short, a notion introduced by P. M. Cohn are studied. Properties and examples of GE-rings and GE_n-rings but not GE-rings are derived. Following the result of Bass, stable rank of a ring R (denoted by sr(R)) is related to the general linear group over R. Every ring with stable rank one is a GE-ring. A principal ideal domain (ring) has stable rank ? 2. For a principal ideal domain R with stable rank one, R must be a Euclidean ring. Examples of GE-rings with stable rank higher than one are given. For the ring of integers O_K in the quadratic field K = Q(√d) with d a square free rational integer, sr(O_K) = 2.

關鍵字(中)

★ GE-ring

關鍵字(英)

★ GE-ring
★ Generalized Euclidean Ring
★ Stable rank
★ Unimodular

論文目次

摘要? i
Abstract ii
Acknowledgement iii
Contents iv
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 GE-rings 5
2.1 GE-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Examples of GE-rings and GEn-rings . . . . . . . . . . . 9
3 Stable Rank of Rings 12
3.1 Stable Rank of Rings . . . . . . . . . . . . . . . . . . . . 12
3.2 Properties of Stable Rank of Rings . . . . . . . . . . . . 15
3.3 Stable Rank of Principal Ideal Rings . . . . . . . . . . . 21
4 Stable Rank One and Stable Rank Higher Than One 24
4.1 Rings with Stable Rank One . . . . . . . . . . . . . . . . 24
4.2 Rings with Stable Rank Higher Than One . . . . . . . . 30
References 33

參考文獻

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指導教授

呂明光(Ming-Guang Leu)

審核日期

2017-1-16

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