| 摘要: | 令 R 代表一個 ring with unity，GL(R, n) 代表 the group of all invertible n by n matrices over R，GE(R, n) 代表 GL(R, n) 的子群，generated by invertible elementary n by n matrices over R。在 1966 的一篇論文裡，P. M. Cohn 稱 R 是一個 generalized Euclidean ring, 或簡稱 GE-ring, 假如, 對每一個正整數 n, GL(R, n)=GE(R, n)。在這次申請的計劃裡，我們將嘗試證明：The ring of algebraic integers in a number field F is a GE-ring, provided that the group of units is infinite.過程中 the stable rank of a ring R 這個概念 及 相關的性質會扮演關鍵性的角色。 一切順利的話，我們應該還會找到更多 GE-rings 的新例子。註：此申請案是本人最想執行的計畫。(第一優先) ;Let R be a ring with unity, GL(R, n) the group of all invertible n by n matrices over R and GE(R, n) the subgroup of GL(R, n) generated by invertible elementary n by n matrices over R. If K is a field, then, in linear algebra, GL(K, n)=GE(K, n) for every positive integer n. If R is a Euclidean ring, then it is well-known that GL(R, n)=GE(R, n) for every positive integer n. In 1966 (Publ. Math. IHES 30 (1966), 5-53), P. M. Cohn introduced the concept of a generalized Euclidean ring, i.e., a ring R with unity is called a generalized Euclidean ring, or GE-ring for short, if and only if GL(R, n)=GE(R, n) for every positive integer n. Recently in 2015, we prove that a ring R is a GE-ring if it is a quasi-Euclidean ring which is an another generalization of the concept of a Euclidean ring. (Recall that, in this project, a ring R is a quasi-Euclidean ring, introduced by B. Bougaut (1977) and A. Leutbecher (1978) respectively, if and only if it is a commutative ring with unity and every pair (b, a) of elements in R has a terminating division chain of finite length starting from it, the pair (b, a) with this property is also called a Euclidean pair by A. Alahmadi, S. K. Jain, T. Y. Lam, and A. Leroy (J. Algebra (2014)). As an example, let A be the ring of all algebraic integers in the field of complex numbers. Then A is a quasi-Euclidean ring, but it is not Euclidean.)The notion of the stable rank of a ring R, denoted by sr(R), was introduced by H. Bass in 1964. The results on the stable rank of rings have close relation to the concept of GE-rings. For example, if sr(R) = 1, then R is a GE-ring. As examples, sr(R) = 1 for every local ring R and every Artinian ring R. In this project we will attempt to prove the following: The ring of algebraic integers in a number field F is a GE-ring, provided that the group of units is infinite.On the way the notion of the stable rank of a ring R and the related properties will play crucial roles.If everything goes smoothly we will also find more new examples of GE-rings. |
