Euclidean Systems Original Research Article

A Euclidean ring such as the integers is equipped with span algorithm for division with remainder. In non-Euclidean Dedekind domains with cyclic class group, the definition of a Euclidean ideal class generalizes the notion of a Euclidean ring. This generalizes the algorithm for division with remainder. LetKbe a number field and letSbe any finite set of primes ofKwhich contains the infinite primesS∞:. Then for any ring ofS-integers ofK, we define a Euclidean system. This further generalizes the notion of a Euclidean ring and the algorithm for division with remainder. We show that under certain conditions, a ring ofS-integers has a Euclidean system.