Euclidean Rings, Principal Ideal Rings, Noetherian Rings

For computations in the ring of integers \(\mathbb {Z}\) one often uses special properties of this ring such as: \(\mathbb {Z}\) is a principal ideal domain, there is division with remainder, there is a unique prime factor decomposition, there is a greatest common divisor, \(\ldots\) Similar properties can be found in the ring K[X] for a field K. In the next two chapters, we will examine such (wishfull) properties that rings can have, abstractly – and also see examples of rings that do not posess these properties.