Euclidean Ring

An integral domain $$R$$ is said to be a Euclidean ring if for every $$a \ne 0$$ in $$R$$ there is defined a non-negative integer, to be denoted by $$d\left( a \right)$$, such that:

(i) For all $$a,b \in R$$, both non-zero, $$d\left( a \right) \leqslant d\left( {ab} \right)$$,

(ii) For any $$a,b \in R$$, both non-zero, there exist $$q,r \in R$$ such that $$a = qb + r$$ when either $$r = 0$$ or $$d\left( r \right) < d\left( b \right)$$.

NoteThe set of integers $$\mathbb{Z}$$ depends on the property of division algorithm. This property is also known as the Euclidean algorithm, which is used to find the greatest common divisors.  This property is mostly satisfied for rings, and as such we can say that such type of rings are called Euclidean rings.