臺灣博碩士論文加值系統

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。

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 finitely 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.

1 Introduction 1
1.1 Introduction 1
1.2 Preliminary 2

2 Group Rings 6
2.1 Denition of Group Rings and Augmentation Mappings 6
2.2 Generalized Euclidean Group Rings 9

3 Integral Group Rings 11
3.1 ZG Is Not a Principal Ideal Ring 11
3.2 Number of Generators of an Ideal in ZCm 15

4 Stable Rank of a Semilocal Ring 23
4.1 Stable Rank of a Ring 23
4.2 Wedderburn-Artin Theorem 27
4.3 Stable Rank of a Semilocal Ring Is One 29

Appendix to Section 3.2 : An Elementary Proof for the Case ZC3 33

References 37

[1] H. Bass, K-theory and stable algebra, Inst. Hautes Etudes Sci. Publ. Math. 22 (1964), 5-60.
[2] H. Bass, Algebraic K-Theory, Benjamin, New York, 1968.
[3] W.-Y. Chang, On a paper of P. M. Cohn, Master thesis, National Central University, Chung-li, Taiwan, 2015.
[4] W.-Y. Chang, C.-R. Cheng, and M.-G. Leu, A remark on the ring of algebraic integers in Q(√