A remark on the ring of algebraic integers in \(\mathbb{Q}(\sqrt {- d})\)
Let \(D\) be an integral domain. If \(a,b\in D\) and there exist elements \(q_1,r_1,q_2,r_2,\dots,r_{n-1},q_n\in D\) with \[ a=q_1b+r_1,b=q_2r+1+r_2,\dots,r_i=q_{i+2}r_{i+1}+r_i,\dots, r_{n-2}=q_nr_{n-1}, \] then one says that the pair \((a,b)\) has a terminating division chain. If this happens for all \(a,b\in D\), then \(D\) is called a quasi-Euclidean domain. It has been shown by \textit{O. T. O'Meara} [J. Reine Angew. Math. 217, 79--108 (1965; Zbl 0128.25502)] that \(D\) is quasi-Euclidean if and only if \(D\) is a \(GE_2\)-ring and a Bezout domain. The authors prove (Theorem 3.2) that every pair \(a,b\in D\) generating \(D\) has such a chain if and only if \(D\) is a \(GE_2\)-ring, and for every non-Euclidean ring of quadratic integers which is a unique factorization domain (i.e., rings of integers of \(\mathbb Q(\sqrt{-d})\) with \(d=19, 43, 67, 163\)) they construct pairs of elements for which there is no finite division chain.
scientific article; zbMATH DE number 6673865