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Euclidean domain

In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used.

More precisely, a Euclidean domain is an integral domain D for which can be defined a function v mapping nonzero elements of D to non-negative 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.