On Generalized Euclidean Rings

Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/73040

Title:	On Generalized Euclidean Rings
Authors:	湯何曼;Tambunan, Abdul Rahman
Contributors:	數學系
Keywords:	GE-ring;GE-ring;Generalized Euclidean Ring;Stable rank;Unimodular
Date:	2017-01-16
Issue Date:	2017-05-05 17:41:44 (UTC+8)
Publisher:	國立中央大學
Abstract:	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。;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.
Appears in Collections:	[Graduate Institute of Mathematics] Electronic Thesis & Dissertation

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