Euclidean-like Characterizations of Dedekind, Krull, and Factorial Domains Original Research Article

Though Euclidean domains are principal ideal domains, the converse is known to be false. We develop a notion like that of the Euclidean ring for which the converse is true. We similarly give new characterizations of Dedekind, Krull, and unique factorization domains. We also introduce the idea of inductive ideal classes and prove results analogous to those obtained by Lenstra for Euclidean ideal classes.