Motzkin 在 1949 年時提出一個判別整環(integral domain)是否為歐幾里德整環(Euclidean domain)的方法。在我最近的一篇論文裡(Osaka J. Math. 45 (2008), 807-818)，針對 restricted Nagata’s pairwise算則，我們也得到一個類似的判別定理可以用來判別一個 unique factorization domain 是否可以在上面定義 restricted Nagata’s pairwise算則。幸運的，最近我們又導出一個類似於上述的判別定理，它可以用來判別一個整環是否可以在上面定義2-stage Euclidean算則。利用這個新的定理，我們成功回答了一個由 G. Cooke (J. reine angew. Math. 282 (1976)) 提出的問題。在這個新的計畫裡，我們會利用這三個類似的內在判別定理來探討Euclidean 算則、 restricted Nagata’s pairwise算則及 2-stage Euclidean算則之間的關係及差異性。我們會去尋找一般性的整環滿足在上面可以定義 ω-stage Euclidean算則但不可以定義 2-stage Euclidean算則，也會去尋找更多非 unique factorization domain 的整環在上面是可以定義2-stage Euclidean算則。我們也會試圖去擴展restricted Nagata’s pairwise算則及 2-stage Euclidean算則的應用面。 ; In 1949, Motzkin gave a characterization of Euclidean domains. Recently in a paper (Osaka J. Math. 45 (2008), 807-818) we obtain an analogous characterization of rings admitting restricted Nagata’s pairwise algorithm. As to 2-stage Euclidean algorithm, very lucky, we also obtain an analogous characterization of 2-stage Euclidean domains. Using this “internal” characterization of 2-stage Euclidean domains we successfully answer a problem raised by G. Cooke (J. reine angew. Math. 282 (1976)). In this new project we will study the relations among Euclidean algorithms, restricted Nagata’s pairwise algorithms and 2-stage Euclidean algorithms by means of these three “internal” characterizations. We will search for more integral domains such that they are -stage Euclidean domains but not 2-stage Euclidean domains. We will also search for more integral domains which are not unique factorization domains but they are 2-stage Euclidean domains. On the way of running this project we will also try to find applications for restricted Nagata’s pairwise algorithms and 2-stage Euclidean algorithms ; 研究期間 9808 ~ 9907
