Editing Euclidean domain (section)

=== Notes on the definition ===

Many authors use other terms in place of "Euclidean function", such as "degree function", "valuation function", "gauge function" or "norm function".<ref name="DummitAlgebra">{{Cite book|title=Abstract Algebra|last1=Dummit|first1=David S.|last2=Foote|first2=Richard M.|publisher=Wiley|year=2004|isbn=9780471433347 |page=270}}</ref> Some authors also require the [[domain of a function|domain]] of the Euclidean function to be the entire ring {{mvar|R}};<ref name="DummitAlgebra"/> however, this does not essentially affect the definition, since (EF1) does not involve the value of {{math|''f''&thinsp;(0)}}. The definition is sometimes generalized by allowing the Euclidean function to take its values in any [[well-ordered set]]; this weakening does not affect the most important implications of the Euclidean property.

The property (EF1) can be restated as follows: for any principal ideal {{mvar|I}} of {{mvar|R}} with nonzero generator {{mvar|b}}, all nonzero classes of the [[quotient ring]] {{math|''R''/''I''}} have a representative {{mvar|r}} with {{math|''f''&thinsp;(''r'') < ''f''&thinsp;(''b'')}}. Since the possible values of {{mvar|f}} are well-ordered, this property can be established by showing that {{math|''f''&thinsp;(''r'') < ''f''&thinsp;(''b'')}} for any {{math|''r'' ∉ ''I''}} with minimal value of {{math|''f''&thinsp;(''r'')}} in its class. Note that, for a Euclidean function that is so established, there need not exist an effective method to determine {{mvar|q}} and {{mvar|r}} in (EF1).