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 integer \mathbb{Z} that depends on the property of division algorithm. This property is also known as the Euclidean algorithm which is use to find the greatest common divisors.  This property is mostly satisfies for rings and for that we can say that such type of rings are called the Euclidean Ring.