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The purpose of this talk 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 !-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 !-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 ring with certain property. Also we will derive an analog of Samuel’s proposition for the concept of 2-stage Euclidean rings, and use it to estimate the smallest 2-stage Euclidean algorithm in global fields. |
The purpose of this talk 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 !-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 !-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 ring with certain property. Also we will derive an analog of Samuel’s proposition for the concept of 2-stage Euclidean rings, and use it to estimate the smallest 2-stage Euclidean algorithm in global fields.