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

- if
*a,b* are nonzero elements of *R* then
d(ab)>=d(a)
- 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)*

__Examples__