 Many authors use other terms in place of "Euclidean function", such as "degree function", "valuation function", "gauge function" or "norm function".Some authors also require the domain of the Euclidean function to be the entire ring ; however this does not essentially affect the definition, since (EF1) does not involve the value of .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.