Euclidean rings of algebraic numbers and functions (Q1223443)

Let \(A\) be an integral domain and \(\psi\) be a map of the non-zero elements of \(A\) into the set \(\mathbb N\) of non-negative integers. Then, \(A\) is said to be Euclidean with respect to the algorithm \(\psi\) if the map \(\psi\) satisfies the condition that for given \(a, b\ne 0\) in \(A\) there are \(c\) and \(d\) in \(A\) such that \(a = bc +d\) and either \(d=0\) or \(\psi(d)< \psi(b)\). In this paper, the author gives some criteria which are useful to show that an integral domain is Euclidean with respect to a multiplicative function. Namely, let \(K\) be either a finite extension of rational number field \(\mathbb R\) or a function field of one variable over any exact constant field \(k\). Let \(S\) be a finite non-empty subset of the set \(X\), which consists of all primes in \(K\), containing all archimedean primes. Denote by \(\mathfrak D_{\mathfrak p}\) the valuation ring of a non-archimedean prime \(\mathfrak p\) of \(K\) and put \(\mathfrak D(X - S) = \cap_{\mathfrak p\in X - S}\mathfrak D_{\mathfrak p}\). Then, he obtains the following results and others by employing an elementary method of approximations: Theorem 2.3. Let \(K\) be a function field of one variable over any exact constant field \(k\). Let \(X\) be the Riemann surface of \(K\) over \(k\). Then, there exists an explicitly computable constant integer \(c>0\) and a non-empty set \(S(c) = \{p\text{ in }X: d(p)\le c\}\) such that the ring \(\mathfrak D(X - S(c))\) is Euclidean with respect to the degree function \(d\). Theorem 3.3. Let \(K\) be a number field. Then there exists an explicitly computable integer \(c>0\) such that if \(S(c)\) contains all primes of \(K\) of norm \(\le c\) and all archimedean primes, then \(\mathfrak D(X - S(c))\) is Euclidean with respect to the norm. Similar results are obtained, respectively, by \textit{O. T. O'Meara} [J. Reine Angew. Math. 217, 79--108 (1965; Zbl 0128.25502)] and by \textit{C. S. Queen} [Bull. Am. Math. Soc. 79, 437--439 (1973; Zbl 0261.12001)] by a different method.