 Euclidean domain

A Euclidean domain is an integral domain $R$ with a function $\delta :R\setminus \{0\}\to \mathbb{N}$ such that, for $a,b\in R$ with $b\ne 0$, there are $q,r\in R$ satisfying $a=bq+r$ and either $r=0$ or $\delta (r)<\delta (b)$. The function need not be an ordinary absolute value.

The integers use $\delta (b)=|b|$ and $k[X]$ over a field uses polynomial degree. Repeated division gives the Euclidean algorithm and a greatest common divisor. Choosing a nonzero element of least Euclidean value in an ideal proves that every ideal is principal, so every Euclidean domain is a principal ideal domain.

A Euclidean domain is a type of integral domain (a non-zero commutative ring with no zero divisors) that satisfies a certain property similar to the division algorithm in the integers.