Let R be a commutative integral domain. We say that R is a Euclidean ring (or ER) if there exists a function d: R\{0} ->N (to the natural numbers) (usually called a norm such that
  1. if a,b are nonzero elements of R then d(ab)>=d(a)
  2. Let a,b in R with a nonzero. Then there exists q,r in R such that b=qa+r with either r=0 or d(r)<d(a)