# 關於 stable rank of rings, GE-rings 及 quasi-Euclidean rings 的研究 On stable rank of rings, GE-rings and quasi-Euclidean rings

## 專案詳細資料

### Description

令 R 代表一個 ring with unity，GL (R) n 代表 the group of all invertible n by n matricesover R，GE (R) n 代表 the subgroup of GL (R) n generated by invertible elementary n by n matricesover R。令 F 代表一個 field，在線性代數裡對應每一個正整數n 我們知道GL (F) GE (F) n n 。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 everypositive integer n 這樣的相等關係。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 R withunity is called a generalized Euclidean ring，or GE-ring for short，if and only if GL (R) n = GE (R) nfor every positive integer n。最近在2015 年，我們證明了一個相關的結果，即 a ring R is aGE-ring if it is a quasi-Euclidean ring。(關於quasi-Euclidean ring 的定義請參考 (二) 計畫英文摘要。)在1964 年，H. Bass introduced the notion of the stable rank of a ring R，denoted by sr(R)。對我們而言，注意到 the stable rank of a ring R 這一個概念，是一個重要的新發現。The results onthe stable rank of rings have close relation to the concepts of GE-rings and n GE -rings。例如，if sr(R)= 1，then R is a GE-ring。實際例子有，sr(R) = 1 for every local ring R and every Artinian ring R。在這次申請的計劃裡，我們將一方面學習、研究 the relations between the stable rank of ringsand the concepts of GE- rings and n GE -rings。同時另一方面希望能找到更多例子，它們是quasi-Euclidean rings 但不是 Euclidean rings，它們是 GE-rings 但不是 quasi-Euclidean rings，也希望能找到更多例子，它們是n GE -rings for some integer n but not GE-rings。(Recall that a ringR with unity is called a n GE -ring if GL (R) GE (R) n n for positive integer n。)註：此申請案是本人最想執行的計畫。(第一優先)
狀態 已完成 1/08/17 → 31/07/18