本論文的主要目的在研究k階歐幾里得環(k-stage Euclidean ring)，它是歐幾里得環的一種推廣。正如同歐幾里得環一樣，我們給出2階歐幾里得環的一種內在判別法。使用此判別法我們能夠提供無窮多個整域(integral domains)是ω階歐幾里得環，但不是2階歐幾里得環。我們所研究的例子解決了一個由 G. E. Cooke 在 [J. reine angew. Math. 282 (1976), 133-156] 所提出的重要問題。此問題敘述如下: “我並不知道是否有一個例子是ω階歐幾里得環，但不是2階歐幾里得環”。此外我們將研究一個 P. Samuel所提出與歐幾里得整域有關的命題，並且對這命題給予兩種版本的推廣。首先我們將這命題推廣到滿足某些條件的環，接著針對2階歐幾里得環我們將給予一個類似於Samuel命題的對應版本，並且給予在整體(global fields)上的應用。 The purpose of this thesis is to investigate the concept of a k-stage Euclidean ring, which is a generalization of the concept of a Euclidean ring. As with Euclidean rings, we will give an internal characterization of 2-stage Euclidean rings. Applying this characterization we are capable of providing infinitely many integral domains, which are ω-stage Euclidean but not 2-stage Euclidean. Our examples solve finally a fundamental question related to the notion of k-stage Euclidean rings raised by G. E. Cooke [J. reine angew. Math. 282 (1976), 133-156]. The question was stated as follows: ``I do not know of an example of an ω-stage euclidean ring which is not 2-stage euclidean." In addition we will study a proposition of P. Samuel, which is related to the concept of Euclidean domains. First we will extend Samuel's result to the commutative rings with certain property. Also we will derive an analog of Samuel's proposition for the concept of 2-stage Euclidean rings, and give application to global fields.
