Euclidean systems
In 1979, \textit{H. W. Lenstra} [Astérisque 61, 121--131 (1979; Zbl 0401.12005)] introduced the notion of a Euclidean ideal class for Dedekind domains \(R\). He showed that \(R\) is Euclidean if and only if the ideal class \([R]\) is, and that any Euclidean ideal class generates the class group; in particular, rings with noncyclic class group do not have a Euclidean ideal class. In the present paper the author generalizes Lenstra's approach by defining Euclidean systems of ideal classes and showing that any Euclidean system generates the class group. Explicit examples of a Euclidean system with a noncyclic class group are apparently not known, but the author's main theorem shows that, under GRH, these should not be too rare: let \(S\) be a set of suitably chosen generators of the class group of a Dedekind domain \(R\) containing the maximal order of a number field \(K\), let \(n\) and \(s\) denote the rank of the class group of \(R\) and the \(\mathbb Z\)-rank of the unit group of \(R\), respectively. Assuming the truth of a certain subset of the Generalized Riemann Hypothesis, there is a Euclidean function that makes \(S\) into a Euclidean system for \(R\) if \(s \geq \max \{1, n-1\}\). This is a generalization of a result of \textit{P. Weinberger} [Proc. Sympos. Pure Math. 24, 321--332 (1973; Zbl 0287.12012)] for the usual Euclidean rings.
scientific article; zbMATH DE number 1247923
scientific article