More precisely, a Euclidean domain is an integral domain D for which can be defined a function v mapping nonzero elements of D to nonnegative integers and possessing the following properties:
The function v is variously called a gauge, valuation or norm. Note that some authors define the function in an inequivalent way which nonetheless still gives the same class of rings.
Examples of Euclidean domains include:
Every Euclidean domain is a principal ideal domain. In fact, if I is a nonzero ideal of a Euclidean domain D and a nonzero a in I is chosen to minimize g(a), then I = aD.
The name comes from the fact that the extended Euclidean algorithm can be carried out in any Euclidean domain.
Search Encyclopedia

Featured Article
