關於 generalized Euclidean rings 及 quasi-Euclidean rings 的研究

在線性代數裡，我們知道每一個invertible n by n matrix over a field F 都可以分解成有限多個elementary matrices over F 相乘。但是，針對任意一個 invertible n by n matrix over an arbitrary ringwith unity 這個性質不一定成立。令 R 代表一個 ring with unity，GL (R) n 代表 the group of allinvertible n by n matrices over R，GE (R) n 代表 the subgroup of GL (R) n generated by invertibleelementary n by n matrices over R。當 R 是 Euclidean ring 時，我們可以在很早期的文獻裡找到GL (R) n = GE (R) n for every positive integer n。在1966年的一篇論文裡(Publ. Math. IHES 30(1966), 5-53)，P. M. Cohn introduced the concept of a generalized Euclidean ring，也就是 a ring Rwith 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。最近，我們證明了一個相關的結果，即 a ring R is a GE-ringif it is a quasi-Euclidean ring which is an another generalization of the concept of a Euclidean ring。(Recall that a ring R is a quasi-Euclidean ring, introduced in 1976, if and only if it is a commutativering with unity and every pair (b, a) of elements in R has a terminating division chain of finite lengthstarting from it, in other words, a greatest common divisor of the pair (b, a) exists in R and it can beobtained by applying a terminating division chain of finite length starting from (b, a).)目前我們是知道一些例子，它們是 quasi-Euclidean rings 但不是 Euclidean rings，也知到一些例子，它們是 GE-rings 但不是 quasi-Euclidean rings。老實說這兩方面的例子在文獻上知道的並不豐富，這也就是吸引人的原因。在我的見識裡，the theories of generalized Euclidean rings andquasi-Euclidean rings，as part of the theory of Euclidean rings，are still underdeveloped。 在這次的計劃裡，我們希望能找到更多例子，它們是 quasi-Euclidean rings 但不是 Euclidean rings，也希望能找到一些例子，它們是 GE-rings 但不是 quasi-Euclidean rings。註：此申請案是本人最想執行的計畫。(第一優先)