Some examples of principal ideal domain which are not Euclidean and some other counterexamples

Veselin Perić, Mirjana Vuković

It is well known that every Euclidean ring is a principal ideal ring. It is also known for a very long time that the converse is not valid. Counterexamples exist under the rings $\mathrm R$ of integral algebraic numbers in quadratic complex fields $\mathbb Q[\sqrt{-D}]$, for $D=19,43,67$, and 163. In conection with these counterexamples several results were published in an effort to make them somewhat more accessible. The aim of this note is to present and complete these results.