### 博碩士論文 972401005 完整後設資料紀錄

 DC 欄位 值 語言 DC.contributor 數學系 zh_TW DC.creator 鄭至人 zh_TW DC.creator Chih-Ren Cheng en_US dc.date.accessioned 2017-7-14T07:39:07Z dc.date.available 2017-7-14T07:39:07Z dc.date.issued 2017 dc.identifier.uri http://ir.lib.ncu.edu.tw:88/thesis/view_etd.asp?URN=972401005 dc.contributor.department 數學系 zh_TW DC.description 國立中央大學 zh_TW DC.description National Central University en_US dc.description.abstract 在1966年，P. M. Cohn 受到佈於歐幾里德環的可逆矩陣可以用基本方陣列簡化為單位矩陣這個性質的啟發，介紹了廣義歐幾里德環的概念。在1984年，Dennis、Magurn 與 Vaserstrin 證明有限循環群Cm的整數群環ZCm是廣義歐幾里德環。已知廣義歐幾里德環是quasi-歐幾里德環且quasi-歐幾里德環是廣義歐幾里德環。本文中，對於非明顯交換群G，我們建構一個ZG的有限生成非主理想環來證明ZG既不是歐幾里德環也不是quasi-歐幾里德環，並且給出ZCm的理想環生成元個數之上界。特別是當m為一個質數的次方時，我們給出更嚴謹的上界。在最後一章裡，藉由Wedderburn-Artin 定理，我們會用一個比Bass的證明更容易理解的方式來證明：半局部環的穩定秩為一，所以它是廣義歐幾里德環。 zh_TW dc.description.abstract In 1966, P. M. Cohn introduced the concept of a generalized Euclidean ring, inspired by the property that any invertible matrix over a Euclidean ring can be row-reduced to the dentity matrix by elementary matrices. In 1984, Dennis, Magurn and Vaserstein proved that the integral group ring ZCm of finite cyclic group Cm is generalized Euclidean. It is well known that a Euclidean ring is quasi-Euclidean and a quasi-Euclidean ring is generalized Euclidean. In this thesis, we construct a fi nitely generated nonprincipal ideal of ZG for nontrivial abelian group G to show that ZG is neither Euclidean nor quasi-Euclidean. Moreover, we give an upper bound for the number of generators of an ideal in ZCm. The case m being a power of a prime is treated more seriously. In the final chapter, following the Wedderburn-Artin Theorem, we give a more accessible proof than Bass′ to show that a semilocal ring has stable rank one, hence it is a generalized Euclidean ring. en_US DC.subject 整數群環 zh_TW DC.subject 廣義歐幾里德 zh_TW DC.subject 半局部環 zh_TW DC.subject 穩定秩 zh_TW DC.subject integral group ring en_US DC.subject generalized Euclidean en_US DC.subject semilocal ring en_US DC.subject stable rank en_US DC.title ZCm 的理想環生成元個數之上限 zh_TW dc.language.iso zh-TW zh-TW DC.title An Upper Bound for the Number of Generators of an Ideal in ZCm en_US DC.type 博碩士論文 zh_TW DC.type thesis en_US DC.publisher National Central University en_US